Solve quadratic equations in standard form with ease using our Quadratic Formula Calculator. Instantly find roots (solutions), vertex, discriminant, and graph for any quadratic equation. Perfect for students, teachers, and engineers who need quick, accurate results.
Enter the coefficients a, b, and c for the quadratic equation ax² + bx + c = 0.
Click Calculate to get:
See your quadratic equation plotted automatically. The graph displays the parabola, vertex, and points of intersection with the x-axis.
This calculator solves any quadratic equation in standard form ax² + bx + c = 0. Here is exactly how to use it:
Your equation must be in the form ax² + bx + c = 0 before you enter coefficients. If you have something like x² = 5x + 6, subtract 5x and 6 from both sides first to get x² − 5x − 6 = 0, then read off a = 1, b = −5, c = −6.
Type the value of a (the coefficient of x²), b (the coefficient of x), and c (the constant) into their respective fields. You can enter integers, decimals, or negative numbers. Do not enter a = 0, because that would make the equation linear rather than quadratic.
The calculator shows your roots (x₁ and x₂ or complex roots), the vertex form y = a(x − h)² + k, a step-by-step solution, and an interactive parabola graph. If the discriminant is positive, your parabola crosses the x-axis at two points. If it equals zero, it touches once. If it is negative, the parabola does not touch the x-axis and the roots are complex.
The values of x where y = 0. These are the x-intercepts of the parabola. If the discriminant is positive, you get two distinct real roots x₁ and x₂. If it is zero, both roots are the same. If negative, the roots are complex numbers of the form a ± bi.
The equation rewritten as y = a(x − h)² + k, where (h, k) is the vertex. This form makes it easy to read the minimum or maximum point directly, plot the parabola by hand, and understand its transformation from the parent function y = x².
The value b² − 4ac shown in the step-by-step solution tells you the nature of the roots before you finish the calculation. A positive discriminant: two real roots. Zero: one repeated root. Negative: two complex roots.
The interactive graph shows your parabola centered on its vertex. The x-axis crossings (if any) are your real roots. The vertex is the lowest point when a > 0 (U-shape) or the highest point when a < 0 (∩-shape). The axis of symmetry is the vertical line x = h, which passes through the vertex and divides the parabola into two mirror-image halves.
The quadratic formula solves any equation in standard form ax² + bx + c = 0 for x:
x = (−b ± √(b² − 4ac)) / 2a
The formula comes from completing the square on the general quadratic. The ± symbol means the formula produces two values, x₁ and x₂, corresponding to the two roots.
Solve x² − 3x − 4 = 0. Here a = 1, b = −3, c = −4.
The parabola opens upward (a = 1 > 0), crosses the x-axis at x = −1 and x = 4, and has its minimum at (1.5, −6.25).
Solve x² − 6x + 9 = 0. Here a = 1, b = −6, c = 9.
This equation factors as (x − 3)² = 0. The parabola is tangent to the x-axis at x = 3.
Solve x² + 2x + 5 = 0. Here a = 1, b = 2, c = 5.
The parabola sits entirely above the x-axis (vertex at (−1, 4)) and never crosses it, so there are no real solutions.
A ball is thrown upward with an initial velocity of 20 m/s from a height of 1 m. Its height is h(t) = −5t² + 20t + 1. To find when it hits the ground, set h(t) = 0: use a = −5, b = 20, c = 1. The positive root gives the landing time (approximately t ≈ 4.05 seconds).
A farmer has 120 m of fence for a rectangular enclosure. If the width is x, then length = (120 − 2x)/2 = 60 − x and area = x(60 − x) = −x² + 60x. To find when the area equals 800 m², set −x² + 60x − 800 = 0. Use a = −1, b = 60, c = −800 → x = 20 or x = 40.
A company's profit function is P(x) = −2x² + 100x − 900, where x is units sold. To find the break-even points (P = 0), enter a = −2, b = 100, c = −900. Roots x = 15 and x = 35 are the two production levels where the business neither profits nor loses.
Satellite dishes, car headlights, and solar concentrators use parabolic shapes because a parabola focuses parallel rays to a single point — the focus. The focus of y = ax² is located at (0, 1/4a). For a dish with a = 0.1, the focus is at height 1/0.4 = 2.5 units from the vertex.
If your equation is 3x² = 7x − 2, rearrange to 3x² − 7x + 2 = 0 before entering coefficients: a = 3, b = −7, c = 2. This gives x = (7 ± √(49 − 24)) / 6 = (7 ± 5) / 6, so x = 2 or x = 1/3.
After you get roots x₁ and x₂, you can verify them using Vieta's formulas: the sum of roots x₁ + x₂ = −b/a, and the product x₁ × x₂ = c/a. For x² − 3x − 4 = 0 with roots 4 and −1: sum = 3 = −(−3)/1 ✓; product = −4 = −4/1 ✓.
The quadratic formula always works, but two other methods — factoring and completing the square — are faster in specific cases and are commonly tested in algebra courses.
Works for any quadratic equation. Enter a, b, and c into this calculator for instant step-by-step results. Use this method when the equation does not factor easily or when decimal roots are acceptable.
Look for two numbers that multiply to ac and add to b. If the equation x² − 5x + 6 = 0 has ac = 6 and b = −5, the numbers −2 and −3 work: x² − 2x − 3x + 6 = (x − 2)(x − 3) = 0, giving x = 2 and x = 3. Factoring only works when roots are rational — if the discriminant is not a perfect square, you cannot factor over the integers.
To solve x² + 6x + 5 = 0 by completing the square:
This process, applied to the general form ax² + bx + c = 0, is exactly how the quadratic formula is derived. Understanding it deepens your insight into why the formula works.
The most frequent mistakes students make with the quadratic formula:
Quickly calculate the sales tax on any purchase amount.
Perform trigonometric, exponential, and logarithmic calculations.
Find volumes of spheres, cylinders, and other 3D shapes.
Perform operations with fractions and mixed numbers.
Compute area for circles, triangles, and rectangles.
Plan loans, investments, and retirement.
The quadratic formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), used to find the roots of any quadratic equation.
The discriminant (b² - 4ac) indicates root types:
Use ( h = -b / 2a ) and ( k = f(h) ) to find the vertex (h, k).
The coefficient “a” controls the parabola’s direction and width.
The equation becomes linear, not quadratic.
Check the discriminant (b² - 4ac). If it’s ≥ 0, your equation has real roots.
Yes, if the discriminant is negative, roots will include imaginary numbers (i).
It shows both the numeric roots and the vertex form, with an interactive parabola graph.
Absolutely. Enter decimal or fractional coefficients (e.g., 0.5, ¾), and the tool will compute precise results.