Solve a quadratic equation with steps: discriminant, roots, vertex and factored form. Runs on your device.
To solve ax² + bx + c = 0, put a, b and c into x = (−b ± √(b² − 4ac)) ÷ 2a, which gives the two values of x. The part under the root, D = b² − 4ac, is the discriminant. For 2x² − 3x − 2 = 0 the page opens with a = 2, b = −3 and c = −2, so D = (−3)² − 4 × 2 × (−2) = 9 + 16 = 25 and x = (3 ± 5) ÷ 4. The two roots are 2 and −1/2. The box How this was worked out repeats each of these lines for your own numbers. Write the equation with the right side equal to 0 first, then read off a, b and c with their signs, and use 0 for a term that is missing: x² − 4 = 0 has b = 0.
If D is above 0 there are two different real roots. If D is exactly 0 there is one repeated real root, −b ÷ 2a, and the parabola only touches the x axis. If D is below 0 there are no real roots, only two complex ones of the form p ± qi, where p = −b ÷ 2a and q = √(−D) ÷ 2|a|. For x² + 2x + 5 = 0, D = 4 − 20 = −16, so the roots are −1 + 2i and −1 − 2i. Complex roots always come as a pair like this when a, b and c are real numbers.
Coefficients can have up to 4 decimals and are scaled to whole numbers inside the page, so the discriminant is exact. When D is a perfect square the roots are rational and are shown as exact fractions with their decimal value, and the page writes the factored form: x² − 5x + 6 has D = 1, roots 3 and 2, and factors as (x − 3)(x − 2). 2x² − 3x − 2 factors as (x − 2)(2x + 1). When D is not a perfect square the roots are irrational, such as 1 ± √2 for x² − 2x − 1, and they are shown rounded to 6 decimals with no whole-number factoring. The vertex is at x = −b ÷ 2a, with y found by putting that x back into the equation. The axis of symmetry is the vertical line through it. It is the lowest point when a is positive and the highest when a is negative.
The coefficient a cannot be 0, because then the equation is a straight line with one solution, and the page shows a message on the a box instead of a result. Each coefficient must be between −1,000,000 and 1,000,000 with at most 4 decimals. Irrational roots use the form x₂ = c ÷ (a × x₁) for the smaller one, so a very large b does not lose digits to subtraction. The page does not solve cubic or higher equations, systems of equations, or inequalities, and it does not move terms for you. Roots are rounded to 6 decimals only when shown.
x = (−b ± √(b² − 4ac)) ÷ 2a for an equation written as ax² + bx + c = 0. For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6, so x = (5 ± √1) ÷ 2 and the roots are 3 and 2.
Take the square root of the positive number −D and attach i. For x² + 2x + 5 = 0, D = −16, so √(−16) = 4i and x = (−2 ± 4i) ÷ 2 = −1 ± 2i. The page shows both complex roots and says there are no real ones.
With a = 0 the x² term disappears and bx + c = 0 is a line with one solution, x = −c ÷ b. The quadratic formula would divide by 2a = 0. Type a number other than 0 in the a box, such as 1 for x² + 3x + 2 = 0.
The x value is −b ÷ 2a and the y value is c − b² ÷ 4a. For 2x² − 3x − 2, x = 3/4 and y = −25/8, which is −3.125. Because a is positive, the vertex is the lowest point.
When the discriminant b² − 4ac is a perfect square, such as 1, 4, 9 or 25. Then the roots are rational and the page prints the factored form, for example (x − 2)(2x + 1). For D = 8 the roots are irrational and there is no such factoring.
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