{"id":27933,"date":"2025-06-20T07:59:32","date_gmt":"2025-06-20T07:59:32","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=27933"},"modified":"2025-06-20T07:59:34","modified_gmt":"2025-06-20T07:59:34","slug":"what-are-the-solutions-of-x2-4x50","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/what-are-the-solutions-of-x2-4x50\/","title":{"rendered":"what are the solutions of x2-4x+5=0"},"content":{"rendered":"\n<p>what are the solutions of x2-4x+5=0<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s solve the quadratic equation:<\/p>\n\n\n\n<p>x2\u22124x+5=0x^2 &#8211; 4x + 5 = 0<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Identify coefficients<\/h3>\n\n\n\n<p>The quadratic equation is in standard form:<\/p>\n\n\n\n<p>ax2+bx+c=0ax^2 + bx + c = 0<\/p>\n\n\n\n<p>Here,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a=1a = 1<\/li>\n\n\n\n<li>b=\u22124b = -4<\/li>\n\n\n\n<li>c=5c = 5<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the discriminant<\/h3>\n\n\n\n<p>The discriminant \u0394\\Delta is given by: \u0394=b2\u22124ac\\Delta = b^2 &#8211; 4ac<\/p>\n\n\n\n<p>Substitute the values: \u0394=(\u22124)2\u22124(1)(5)=16\u221220=\u22124\\Delta = (-4)^2 &#8211; 4(1)(5) = 16 &#8211; 20 = -4<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Analyze the discriminant<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If \u0394>0\\Delta > 0, two real and distinct solutions.<\/li>\n\n\n\n<li>If \u0394=0\\Delta = 0, one real solution (a repeated root).<\/li>\n\n\n\n<li>If \u0394&lt;0\\Delta &lt; 0, two complex conjugate solutions.<\/li>\n<\/ul>\n\n\n\n<p>Since \u0394=\u22124\\Delta = -4 (less than zero), the equation has <strong>two complex conjugate solutions<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Use the quadratic formula<\/h3>\n\n\n\n<p>The quadratic formula is: x=\u2212b\u00b1\u03942ax = \\frac{-b \\pm \\sqrt{\\Delta}}{2a}<\/p>\n\n\n\n<p>Since \u0394=\u22124\\Delta = -4, we write: x=\u2212(\u22124)\u00b1\u221242\u00d71=4\u00b1\u221242x = \\frac{-(-4) \\pm \\sqrt{-4}}{2 \\times 1} = \\frac{4 \\pm \\sqrt{-4}}{2}<\/p>\n\n\n\n<p>Recall that \u22124=4\u00d7\u22121=2i\\sqrt{-4} = \\sqrt{4} \\times \\sqrt{-1} = 2i, where ii is the imaginary unit (i2=\u22121i^2 = -1).<\/p>\n\n\n\n<p>So, x=4\u00b12i2=2\u00b1ix = \\frac{4 \\pm 2i}{2} = 2 \\pm i<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Final solutions:<\/strong><\/h3>\n\n\n\n<p>x=2+iandx=2\u2212ix = 2 + i \\quad \\text{and} \\quad x = 2 &#8211; i<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The quadratic equation x2\u22124x+5=0x^2 &#8211; 4x + 5 = 0 has no real solutions because the discriminant is negative. This means the parabola y=x2\u22124x+5y = x^2 &#8211; 4x + 5 does not intersect the x-axis. Instead, its solutions are complex numbers involving the imaginary unit ii.<\/p>\n\n\n\n<p>Complex solutions always come in conjugate pairs when the coefficients of the quadratic are real numbers. The solutions 2+i2 + i and 2\u2212i2 &#8211; i reflect this conjugate pair nature.<\/p>\n\n\n\n<p>The real part of both solutions is 2, indicating a &#8220;center&#8221; of the parabola along the x-axis, while the imaginary part \u00b1i\\pm i represents the distance from the real axis in the complex plane.<\/p>\n\n\n\n<p>Quadratic equations like this often appear in algebra, engineering, and physics when modeling systems where the solutions represent oscillations or waves rather than straightforward real values.<\/p>\n\n\n\n<p>If you graph this quadratic, the curve will open upwards with a vertex at x=2x = 2, y=1y = 1, and never touch the x-axis, which aligns with the fact that it has complex solutions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>what are the solutions of x2-4x+5=0 The correct answer and explanation is: Let&#8217;s solve the quadratic equation: x2\u22124x+5=0x^2 &#8211; 4x + 5 = 0 Step 1: Identify coefficients The quadratic equation is in standard form: ax2+bx+c=0ax^2 + bx + c = 0 Here, Step 2: Calculate the discriminant The discriminant \u0394\\Delta is given by: \u0394=b2\u22124ac\\Delta [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-27933","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27933","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=27933"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27933\/revisions"}],"predecessor-version":[{"id":27934,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27933\/revisions\/27934"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=27933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=27933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=27933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}