{"id":61933,"date":"2021-05-31T08:32:18","date_gmt":"2021-05-31T06:32:18","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61933"},"modified":"2026-10-23T16:29:41","modified_gmt":"2026-10-23T13:29:41","slug":"quadratic-equation-2","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-equation-2\/","title":{"rendered":"Quadratic Formula Calculator \u2013 Solve ax\u00b2 + bx + c = 0"},"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\">Solve Quadratic Equations<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61933\" data-favorite-title=\"Quadratic Equation\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-equation-2\/\" data-favorite-excerpt=\"Solve any quadratic equation ax\u00b2 + bx + c = 0 instantly. Enter the coefficients a, b and c, and the calculator finds both real or complex roots using the quadratic formula and discriminant.\" 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\/quadratic-equation-2\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61933\" data-post=\"61933\"><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=\"text\" name=\"a_coefficient_of_x\" id=\"field-1\" aria-describedby=\"calc-info-1\" placeholder=\"1\" step=\"any\" value=\"1\"\/><small id=\"calc-info-1\">Integer, decimal or fraction, e.g. 2, 0.5 or 3\/4<\/small><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">b (coefficient of x)<\/label><input type=\"text\" name=\"b_coefficient_of_x\" id=\"field-2\" placeholder=\"-3\" step=\"any\" value=\"-3\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">c (constant)<\/label><input type=\"text\" name=\"c_constant\" id=\"field-3\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61933\">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>x\u2081<\/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 class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>x\u2082<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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 class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Discriminant<\/span><span class=\"result-term\">D<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/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 class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Vertex and axis of symmetry<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/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 class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Factored form<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/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 class=\"field-result is-column\" id=\"output-6\"><span class=\"field-title\"><span>Step-by-step solution<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/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 class=\"field-result is-column\" id=\"output-7\"><span class=\"field-title\"><span>Graph of the parabola<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/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=\"61933\" 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>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\\[ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\]\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>\\(D = b^2 - 4ac\\) \u2014 the discriminant, which determines the type of the roots<\/li>\n<\/ul>\n<p>The formula gives the two solutions of the quadratic equation \\(ax^2 + bx + c = 0\\). The discriminant decides what kind of roots you get:<\/p>\n<ul>\n<li>\\(D &gt; 0\\) \u2014 two distinct real roots<\/li>\n<li>\\(D = 0\\) \u2014 one real double root, \\(x_1 = x_2 = -\\frac{b}{2a}\\)<\/li>\n<li>\\(D &lt; 0\\) \u2014 two complex conjugate roots \\(\\frac{-b}{2a} \\pm \\frac{\\sqrt{-D}}{2a}i\\)<\/li>\n<\/ul>\n<p>For a deeper treatment of this term under the square root, see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/discriminant-quadratic-equation\/\">discriminant of a quadratic equation<\/a>.<\/p>\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>Quadratic Equation \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> solve \\(x^2 - 3x + 2 = 0\\), so \\(a = 1\\), \\(b = -3\\), \\(c = 2\\).<\/p>\n<p>\\[ D = (-3)^2 - 4 \\cdot 1 \\cdot 2 = 9 - 8 = 1 \\]<\/p>\n<p>\\[ x_{1,2} = \\frac{-(-3) \\pm \\sqrt{1}}{2 \\cdot 1} = \\frac{3 \\pm 1}{2} \\]<\/p>\n<p>\\[ x_1 = \\frac{3 + 1}{2} = 2, \\quad x_2 = \\frac{3 - 1}{2} = 1 \\]<\/p>\n<p><strong>Result:<\/strong> \\(x_1 = 2.0000\\), \\(x_2 = 1.0000\\). Check: \\(2^2 - 3 \\cdot 2 + 2 = 0\\) and \\(1 - 3 + 2 = 0\\).<\/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>. It must not be zero.<\/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>Read the results in the fields <strong>x\u2081<\/strong> and <strong>x\u2082<\/strong>. Real roots are shown as decimal numbers; complex roots appear in the form \\(p + qi\\) and \\(p - qi\\).<\/li>\n<\/ol>\n<h2>What Is a Quadratic Equation?<\/h2>\n<p>A quadratic equation is any equation that can be written in the standard form \\(ax^2 + bx + c = 0\\), where \\(a \\neq 0\\). The graph of the corresponding function \\(y = ax^2 + bx + c\\) is a parabola: it opens upward when \\(a &gt; 0\\) and downward when \\(a &lt; 0\\). The real roots of the equation are exactly the points where this parabola crosses the x-axis, which is why solving quadratics is the standard way to find intersections, maximum or minimum values, and optimal points in applied problems.<\/p>\n<p>Quadratic equations appear throughout algebra, geometry, physics, engineering and economics \u2014 projectile motion, area and dimension problems, revenue optimization and circuit analysis all reduce to finding roots of a second-degree polynomial. The quadratic formula is guaranteed to work for every quadratic equation, even when factoring is difficult or impossible, which makes it the most reliable solving method.<\/p>\n<p>The nature of the answer is decided by the discriminant \\(D = b^2 - 4ac\\). If the calculator returns two different numbers, the parabola crosses the x-axis twice. If both results are equal, the parabola just touches the axis at its vertex. If the results contain \\(i\\), there are no real solutions: the parabola never reaches the x-axis, and the two roots are <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/\">complex conjugates<\/a> of each other. You can explore such values further with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-number-calculator\/\">complex number calculator<\/a>.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li>The coefficient \\(a\\) must be non-zero. If \\(a = 0\\), the equation is linear, not quadratic, and the calculator reports \"Not a quadratic equation\".<\/li>\n<li>Results are rounded to 4 decimal places. For most practical purposes this is more than enough, but keep rounding in mind if you substitute the roots back into the original equation for verification.<\/li>\n<li>If your equation is not in standard form, rearrange it first: move every term to the left side so that \\(ax^2 + bx + c = 0\\), and only then read off the coefficients. Watch the signs \u2014 entering \\(-3\\) instead of \\(3\\) for \\(b\\) changes the answer completely.<\/li>\n<li>A quick way to check the result: by Vieta's formulas, \\(x_1 + x_2 = -\\frac{b}{a}\\) and \\(x_1 x_2 = \\frac{c}{a}\\). If both hold, the roots are correct.<\/li>\n<li>For higher-degree polynomials, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/cubic-equation\/\">cubic equation solver<\/a> or the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quartic-equation\/\">quartic equation solver<\/a>. If you want to expand a factored form back into standard form first, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/foil-method\/\">FOIL method calculator<\/a> is useful.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Solve any quadratic equation ax\u00b2 + bx + c = 0 instantly. Enter the coefficients a, b and c, and the calculator finds both real or complex roots using the quadratic formula and discriminant.<\/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-61933","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\/61933","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=61933"}],"version-history":[{"count":6,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61933\/revisions"}],"predecessor-version":[{"id":67483,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61933\/revisions\/67483"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}