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<div class="headertitle"><div class="title">quadratic_equations_complex_numbers.cpp</div></div>
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<div class="contents">
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<a href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span></div>
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<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span> </div>
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<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span><span class="preprocessor">#include <array></span> </div>
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<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span><span class="preprocessor">#include <cassert></span> </div>
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<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span><span class="preprocessor">#include <cmath></span> </div>
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<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span><span class="preprocessor">#include <complex></span> </div>
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<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span><span class="preprocessor">#include <exception></span> </div>
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<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="preprocessor">#include <iomanip></span> </div>
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<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span><span class="preprocessor">#include <iostream></span> </div>
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<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span></div>
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<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../dd/d47/namespacemath.html">math</a> {</div>
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<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span></div>
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<div class="foldopen" id="foldopen00053" data-start="{" data-end="}">
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<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"><a class="line" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb"> 53</a></span>std::array<std::complex<long double>, 2> <a class="code hl_function" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">quadraticEquation</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> a,</div>
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<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> b,</div>
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<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> c) {</div>
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<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> <span class="keywordflow">if</span> (a == 0) {</div>
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<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">"quadratic coefficient cannot be 0"</span>);</div>
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<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> }</div>
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<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> </div>
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<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> discriminant = b * b - 4 * a * c;</div>
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<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> std::array<std::complex<long double>, 2> solutions{0, 0};</div>
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<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> </div>
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<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> <span class="keywordflow">if</span> (discriminant == 0) {</div>
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<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> solutions[0] = -b * 0.5 / a;</div>
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<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> solutions[1] = -b * 0.5 / a;</div>
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<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> <span class="keywordflow">return</span> solutions;</div>
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<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> }</div>
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<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> </div>
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<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> <span class="comment">// Complex root (discriminant < 0)</span></div>
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<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> <span class="comment">// Note that the left term (-b / 2a) is always real. The imaginary part</span></div>
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<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> <span class="comment">// appears when b^2 - 4ac < 0, so sqrt(b^2 - 4ac) has no real roots. So,</span></div>
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<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> <span class="comment">// the imaginary component is i * (+/-)sqrt(abs(b^2 - 4ac)) / 2a.</span></div>
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<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span> <span class="keywordflow">if</span> (discriminant > 0) {</div>
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<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> <span class="comment">// Since discriminant > 0, there are only real roots. Therefore,</span></div>
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<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> <span class="comment">// imaginary component = 0.</span></div>
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<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> solutions[0] = std::complex<long double>{</div>
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<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> (-b - std::sqrt(discriminant)) * 0.5 / a, 0};</div>
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<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> solutions[1] = std::complex<long double>{</div>
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<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> (-b + std::sqrt(discriminant)) * 0.5 / a, 0};</div>
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<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <span class="keywordflow">return</span> solutions;</div>
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<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> }</div>
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<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> <span class="comment">// Since b^2 - 4ac is < 0, for faster computation, -discriminant is</span></div>
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<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> <span class="comment">// enough to make it positive.</span></div>
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<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> solutions[0] = std::complex<long double>{</div>
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<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> -b * 0.5 / a, -std::sqrt(-discriminant) * 0.5 / a};</div>
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<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> solutions[1] = std::complex<long double>{</div>
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<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> -b * 0.5 / a, std::sqrt(-discriminant) * 0.5 / a};</div>
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<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> </div>
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<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> <span class="keywordflow">return</span> solutions;</div>
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<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span>}</div>
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</div>
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<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> </div>
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<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span>} <span class="comment">// namespace math</span></div>
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<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span></div>
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<div class="foldopen" id="foldopen00100" data-start="{" data-end="}">
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<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"><a class="line" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#af7a6d4e3dc85a6288c8f1f7094830c5a"> 100</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#af7a6d4e3dc85a6288c8f1f7094830c5a">assertArray</a>(std::array<std::complex<long double>, 2> input,</div>
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<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span> std::array<std::complex<long double>, 2> expected,</div>
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<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span> <span class="keywordtype">size_t</span> precision = 10) {</div>
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<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> exponent = std::pow(10, precision);</div>
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<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> input[0].real(std::round(input[0].real() * exponent));</div>
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<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> input[1].real(std::round(input[1].real() * exponent));</div>
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<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> input[0].imag(std::round(input[0].imag() * exponent));</div>
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<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> input[1].imag(std::round(input[1].imag() * exponent));</div>
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<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> </div>
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<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> expected[0].real(std::round(expected[0].real() * exponent));</div>
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<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> expected[1].real(std::round(expected[1].real() * exponent));</div>
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<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> expected[0].imag(std::round(expected[0].imag() * exponent));</div>
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<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> expected[1].imag(std::round(expected[1].imag() * exponent));</div>
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<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> </div>
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<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> assert(input == expected);</div>
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<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span>}</div>
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</div>
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<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span></div>
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<div class="foldopen" id="foldopen00122" data-start="{" data-end="}">
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<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"><a class="line" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#aa8dca7b867074164d5f45b0f3851269d"> 122</a></span><span class="keyword">static</span> <span class="keywordtype">void</span> <a class="code hl_function" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>() {</div>
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<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> <span class="comment">// Values are equal and real.</span></div>
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<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span> std::cout << <span class="stringliteral">"Input: \n"</span></div>
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<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> <span class="stringliteral">"a=1 \n"</span></div>
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<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> <span class="stringliteral">"b=-2 \n"</span></div>
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<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> <span class="stringliteral">"c=1 \n"</span></div>
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<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> <span class="stringliteral">"Expected output: \n"</span></div>
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<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> <span class="stringliteral">"(1, 0), (1, 0)\n\n"</span>;</div>
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<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span> std::array<std::complex<long double>, 2> equalCase{</div>
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<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> std::complex<long double>{1, 0}, std::complex<long double>{1, 0}};</div>
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<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">math::quadraticEquation</a>(1, -2, 1) == equalCase);</div>
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<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> </div>
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<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> <span class="comment">// Values are equal and complex.</span></div>
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<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> std::cout << <span class="stringliteral">"Input: \n"</span></div>
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<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span> <span class="stringliteral">"a=1 \n"</span></div>
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<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> <span class="stringliteral">"b=4 \n"</span></div>
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<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> <span class="stringliteral">"c=5 \n"</span></div>
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<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> <span class="stringliteral">"Expected output: \n"</span></div>
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<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> <span class="stringliteral">"(-2, -1), (-2, 1)\n\n"</span>;</div>
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<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span> std::array<std::complex<long double>, 2> complexCase{</div>
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<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> std::complex<long double>{-2, -1}, std::complex<long double>{-2, 1}};</div>
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<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">math::quadraticEquation</a>(1, 4, 5) == complexCase);</div>
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<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> </div>
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<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span> <span class="comment">// Values are real.</span></div>
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<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> std::cout << <span class="stringliteral">"Input: \n"</span></div>
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<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> <span class="stringliteral">"a=1 \n"</span></div>
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<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> <span class="stringliteral">"b=5 \n"</span></div>
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<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> <span class="stringliteral">"c=1 \n"</span></div>
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<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span> <span class="stringliteral">"Expected output: \n"</span></div>
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<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span> <span class="stringliteral">"(-4.7912878475, 0), (-0.2087121525, 0)\n\n"</span>;</div>
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<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> std::array<std::complex<long double>, 2> floatCase{</div>
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<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> std::complex<long double>{-4.7912878475, 0},</div>
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<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> std::complex<long double>{-0.2087121525, 0}};</div>
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<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> <a class="code hl_function" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#af7a6d4e3dc85a6288c8f1f7094830c5a">assertArray</a>(<a class="code hl_function" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">math::quadraticEquation</a>(1, 5, 1), floatCase);</div>
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<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> </div>
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<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <span class="comment">// Values are complex.</span></div>
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<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> std::cout << <span class="stringliteral">"Input: \n"</span></div>
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<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="stringliteral">"a=1 \n"</span></div>
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<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <span class="stringliteral">"b=1 \n"</span></div>
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<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> <span class="stringliteral">"c=1 \n"</span></div>
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<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="stringliteral">"Expected output: \n"</span></div>
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<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> <span class="stringliteral">"(-0.5, -0.8660254038), (-0.5, 0.8660254038)\n\n"</span>;</div>
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<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> std::array<std::complex<long double>, 2> ifloatCase{</div>
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<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> std::complex<long double>{-0.5, -0.8660254038},</div>
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<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> std::complex<long double>{-0.5, 0.8660254038}};</div>
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<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> <a class="code hl_function" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#af7a6d4e3dc85a6288c8f1f7094830c5a">assertArray</a>(<a class="code hl_function" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">math::quadraticEquation</a>(1, 1, 1), ifloatCase);</div>
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<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> </div>
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<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> std::cout << <span class="stringliteral">"Exception test: \n"</span></div>
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<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> <span class="stringliteral">"Input: \n"</span></div>
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<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> <span class="stringliteral">"a=0 \n"</span></div>
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<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> <span class="stringliteral">"b=0 \n"</span></div>
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<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> <span class="stringliteral">"c=0\n"</span></div>
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<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> <span class="stringliteral">"Expected output: Exception thrown \n"</span>;</div>
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<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> <span class="keywordflow">try</span> {</div>
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<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> <a class="code hl_function" href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">math::quadraticEquation</a>(0, 0, 0);</div>
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<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> } <span class="keywordflow">catch</span> (std::invalid_argument& e) {</div>
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<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> std::cout << <span class="stringliteral">"Exception thrown successfully \n"</span>;</div>
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<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> }</div>
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<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span>}</div>
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</div>
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<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span></div>
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<div class="foldopen" id="foldopen00186" data-start="{" data-end="}">
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<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"><a class="line" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4"> 186</a></span><span class="keywordtype">int</span> <a class="code hl_function" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a>() {</div>
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<div class="line"><a id="l00187" name="l00187"></a><span class="lineno"> 187</span> <a class="code hl_function" href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// Run self-test implementation.</span></div>
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<div class="line"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"> 189</span>}</div>
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</div>
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<div class="ttc" id="anamespacemath_html"><div class="ttname"><a href="../../dd/d47/namespacemath.html">math</a></div><div class="ttdoc">for assert</div></div>
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<div class="ttc" id="anamespacemath_html_aacb1411ef2029e81f249c21e17c96fdb"><div class="ttname"><a href="../../dd/d47/namespacemath.html#aacb1411ef2029e81f249c21e17c96fdb">math::quadraticEquation</a></div><div class="ttdeci">std::array< std::complex< long double >, 2 > quadraticEquation(long double a, long double b, long double c)</div><div class="ttdoc">Quadratic equation calculator.</div><div class="ttdef"><b>Definition</b> <a href="#l00053">quadratic_equations_complex_numbers.cpp:53</a></div></div>
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<div class="ttc" id="aquadratic__equations__complex__numbers_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementations to test quadraticEquation function.</div><div class="ttdef"><b>Definition</b> <a href="#l00122">quadratic_equations_complex_numbers.cpp:122</a></div></div>
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<div class="ttc" id="aquadratic__equations__complex__numbers_8cpp_html_ae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdoc">Main function.</div><div class="ttdef"><b>Definition</b> <a href="#l00186">quadratic_equations_complex_numbers.cpp:186</a></div></div>
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<div class="ttc" id="aquadratic__equations__complex__numbers_8cpp_html_af7a6d4e3dc85a6288c8f1f7094830c5a"><div class="ttname"><a href="../../da/d18/quadratic__equations__complex__numbers_8cpp.html#af7a6d4e3dc85a6288c8f1f7094830c5a">assertArray</a></div><div class="ttdeci">void assertArray(std::array< std::complex< long double >, 2 > input, std::array< std::complex< long double >, 2 > expected, size_t precision=10)</div><div class="ttdoc">Asserts an array of complex numbers.</div><div class="ttdef"><b>Definition</b> <a href="#l00100">quadratic_equations_complex_numbers.cpp:100</a></div></div>
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