{"id":61498,"date":"2021-02-02T16:49:21","date_gmt":"2021-02-02T14:49:21","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61498"},"modified":"2026-09-30T20:27:02","modified_gmt":"2026-09-30T17:27:02","slug":"discriminant-quadratic-equation","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/discriminant-quadratic-equation\/","title":{"rendered":"Discriminant Quadratic Equation Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Quadratic Discriminant Calculator<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61498\" data-favorite-title=\"Discriminant Quadratic Equation\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/discriminant-quadratic-equation\/\" data-favorite-excerpt=\"Enter the coefficients a, b and c of a quadratic equation ax\u00b2 + bx + c = 0 to instantly compute its discriminant \u0394 = b\u00b2 \u2212 4ac and find out how many real or complex roots the equation has.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/discriminant-quadratic-equation\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61498\" data-post=\"61498\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">a (coefficient of x\u00b2)<\/label><input type=\"number\" name=\"a_coefficient_of_x\" id=\"field-1\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">b (coefficient of x)<\/label><input type=\"number\" name=\"b_coefficient_of_x\" id=\"field-2\" placeholder=\"2\" step=\"1\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">c (constant)<\/label><input type=\"number\" name=\"c_constant\" id=\"field-3\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61498\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Discriminant<\/span><span class=\"result-term\">\u0394<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"61498\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Discriminant of a Quadratic Equation Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\Delta = b^2 - 4ac \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a\\) \u2014 coefficient of \\(x^2\\) (must not be zero)<\/li>\n<li>\\(b\\) \u2014 coefficient of \\(x\\)<\/li>\n<li>\\(c\\) \u2014 constant term<\/li>\n<li>\\(\\Delta\\) (Delta) \u2014 the discriminant of the equation \\(ax^2 + bx + c = 0\\)<\/li>\n<\/ul>\n<p>The discriminant is the expression under the square root in the quadratic formula. Its sign tells you the nature of the roots without solving the equation:<\/p>\n<ul>\n<li>\\(\\Delta &gt; 0\\) \u2014 two distinct real roots<\/li>\n<li>\\(\\Delta = 0\\) \u2014 one real repeated root<\/li>\n<li>\\(\\Delta &lt; 0\\) \u2014 two complex conjugate roots<\/li>\n<\/ul>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Discriminant \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the discriminant of \\(x^2 + 2x + 1 = 0\\), so \\(a = 1\\), \\(b = 2\\), \\(c = 1\\).<\/p>\n<p>\\[ \\Delta = b^2 - 4ac = 2^2 - 4 \\cdot 1 \\cdot 1 = 4 - 4 = 0 \\]<\/p>\n<p><strong>Result:<\/strong> \\(\\Delta = 0\\). The equation has exactly one real repeated root, \\(x = -1\\).<\/p>\n<p><strong>Another example:<\/strong> for \\(x^2 - 3x + 2 = 0\\) with \\(a = 1\\), \\(b = -3\\), \\(c = 2\\):<\/p>\n<p>\\[ \\Delta = (-3)^2 - 4 \\cdot 1 \\cdot 2 = 9 - 8 = 1 \\]<\/p>\n<p><strong>Result:<\/strong> \\(\\Delta = 1 &gt; 0\\), so there are two distinct real roots.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the coefficient of \\(x^2\\) in the field <strong>a (coefficient of x\u00b2)<\/strong>.<\/li>\n<li>Enter the coefficient of \\(x\\) in the field <strong>b (coefficient of x)<\/strong>.<\/li>\n<li>Enter the constant term in the field <strong>c (constant)<\/strong>.<\/li>\n<li>The <strong>Discriminant<\/strong> result updates automatically and shows the value of \\(\\Delta\\).<\/li>\n<\/ol>\n<p>Use any real numbers, including negatives and decimals. Make sure \\(a \\neq 0\\); if \\(a = 0\\), the equation is not quadratic but linear, and the discriminant is not defined.<\/p>\n<h2>What Is the Discriminant?<\/h2>\n<p>Every quadratic equation can be written in the standard form \\(ax^2 + bx + c = 0\\). Its roots are given by the quadratic formula:<\/p>\n<p>\\[ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\]<\/p>\n<p>The expression under the square root, \\(\\Delta = b^2 - 4ac\\), is called the discriminant (from the Latin \\discernere\\, \"to distinguish\"), because it distinguishes between the possible types of roots.<\/p>\n<p>The logic is simple: the formula takes the square root of \\(\\Delta\\). If \\(\\Delta &gt; 0\\), the square root is a positive real number, so the \\(\\pm\\) sign produces two different real roots. If \\(\\Delta = 0\\), both branches of the formula collapse to the single value \\(x = -b\/2a\\), giving a repeated root. If \\(\\Delta &lt; 0\\), the square root of a negative number is not real; the two roots become a pair of complex conjugates with the same real part \\(-b\/2a\\) and opposite imaginary parts.<\/p>\n<p>The discriminant is a quick diagnostic tool: before doing any solving, one subtraction tells you whether the parabola \\(y = ax^2 + bx + c\\) crosses the x-axis twice, touches it once, or never reaches it. This is used everywhere from school algebra to engineering, physics, and optimization, where knowing whether a system has real solutions matters as much as the solutions themselves.<\/p>\n<h2>Interpreting the Result<\/h2>\n<ul>\n<li><strong>\\(\\Delta &gt; 0\\):<\/strong> two distinct real roots, \\(x = \\frac{-b \\pm \\sqrt{\\Delta}}{2a}\\). The parabola crosses the x-axis at two points.<\/li>\n<li><strong>\\(\\Delta = 0\\):<\/strong> one real root of multiplicity two, \\(x = -\\frac{b}{2a}\\). The parabola is tangent to the x-axis at its vertex.<\/li>\n<li><strong>\\(\\Delta &lt; 0\\):<\/strong> no real roots; two complex conjugate roots \\(x = \\frac{-b \\pm i\\sqrt{|\\Delta|}}{2a}\\). The parabola lies entirely above or below the x-axis. You can explore these roots with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-root\/\">Complex Root Calculator<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-number-calculator\/\">Complex Number Calculator<\/a>.<\/li>\n<\/ul>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li>The calculator assumes a genuine quadratic equation, so \\(a \\neq 0\\). If \\(a = 0\\), the equation reduces to \\(bx + c = 0\\) and should be solved as linear.<\/li>\n<li>The result is rounded to three decimal places. With very large coefficients, the exact value of \\(\\Delta\\) may have more digits than shown.<\/li>\n<li>The sign of \\(\\Delta\\), not its magnitude, determines the type of roots. A larger positive \\(\\Delta\\) simply means the two real roots are farther apart.<\/li>\n<li>\\(\\Delta\\) tells you <em>how many<\/em> roots of each type exist, but not their values. To find the roots themselves, apply the quadratic formula or use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-equation-2\/\">Quadratic Equation Calculator<\/a>.<\/li>\n<li>For higher-degree polynomials, different tools are needed \u2014 see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/cubic-equation\/\">Cubic Equation Calculator<\/a> for third-degree equations.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter the coefficients a, b and c of a quadratic equation ax\u00b2 + bx + c = 0 to instantly compute its discriminant \u0394 = b\u00b2 \u2212 4ac and find out how many real or complex roots the equation has.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-61498","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61498","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61498"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61498\/revisions"}],"predecessor-version":[{"id":66227,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61498\/revisions\/66227"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}