Quadratic Equation Calculator

What a quadratic equation means

A quadratic equation has the form ax² + bx + c = 0, where a is not zero. The highest power is 2, so its graph is a parabola rather than a straight line. The coefficients control the shape and position: a changes the opening and width, b shifts the balance of the terms, and c is the y-intercept. The quadratic formula works even when the roots are awkward fractions or decimals, so you do not need to guess factors.

The formula is x = (-b ± √(b² - 4ac)) / (2a). The expression inside the square root is the discriminant, D. This page follows the usual real-number convention and also reports complex roots when D is negative. Entering a = 0 is handled separately because the equation has then become linear.

The current 2026 edition of OpenStax College Algebra 2e covers quadratic equations, factoring, completing the square, and the formula in one section. The OpenStax quadratic equations reference is useful when you want the derivation behind the calculator.

Worked example: 2x^2 - 4x - 6 = 0

Take a = 2, b = -4, and c = -6. The discriminant is D = (-4)² - 4(2)(-6) = 16 + 48 = 64. Since the square root of 64 is 8, the two roots are x = (4 + 8) / 4 = 3 and x = (4 - 8) / 4 = -1. The equation therefore factors as 2(x - 3)(x + 1) = 0.

The vertex provides a second view. Its x-coordinate is -b/(2a) = 4/4 = 1. Substituting 1 gives y = 2(1)² - 4(1) - 6 = -8, so the vertex is (1, -8). The axis of symmetry is x = 1, exactly halfway between -1 and 3. Vieta's rules provide a fast check: the root sum is -b/a = 2, while 3 + (-1) = 2; the product is c/a = -3, while 3(-1) = -3.

Substitution closes the check. At x = 3, 2(9) - 12 - 6 = 0. At x = -1, 2(1) + 4 - 6 = 0. If rounding makes a check look like 0.000001 instead of 0, the displayed roots were rounded while the internal calculation retains more digits.

Completing the square on a messy leading coefficient

Completing the square is the route that produces the quadratic formula, and it still earns its keep when you want vertex form or a hand solution with clean halves. Take 3x² - 12x + 7 = 0, where the leading 3 blocks a direct square. Divide every term by 3 to get x² - 4x + 7/3 = 0, then move the constant: x² - 4x = -7/3. Half of -4 is -2, and squaring it gives 4, so add 4 to both sides and the left side folds into a single square.

The formula agrees. For the same equation, D = 144 - 84 = 60, and (12 +/- sqrt(60)) / 6 gives 3.291 and 0.709. Substitution confirms both: 3(3.291)² - 12(3.291) + 7 = 0 to within rounding, and the same holds at 0.709. The steps also hand you the vertex form for free: 3(x - 2)² - 5 expands back to 3x² - 12x + 7, since at x = 0 it gives 12 - 5 = 7 and at x = 1 it gives 3 - 5 = -2, both matching the original coefficients. The vertex is therefore (2, -5).

Choosing between factoring, the formula, and completing the square

Three methods solve the same equation, and the fastest one depends on the coefficients. Factoring wins when the roots are rational, because two seconds of inspection beats a square root. The formula always works and handles irrational or complex roots without complaint. Completing the square earns its place when vertex form is the goal, as in graphing or optimization problems, since it rewrites the equation as a(x - h)² + k in a few moves.

A practical order: try a quick factor check first, since integer roots near the factors of c show up often in coursework. If nothing appears within a minute, run the formula; it cannot fail while a is nonzero. Switch to completing the square when the question asks for a maximum, a minimum, or a graph's turning point, because the vertex falls out of the rewrite without extra work. The calculator mirrors this workflow by printing the factored form when the roots are rational and the vertex and discriminant in every case, so you can compare methods on the same numbers instead of trusting one path.

The discriminant predicts the roots

D is more than an intermediate number. When D is positive, the parabola crosses the x-axis twice and the calculator returns two different real roots. When D is zero, it touches the axis once and the repeated root is -b/(2a). When D is negative, the graph has no real x-intercept; the conjugate complex roots have real part -b/(2a) and imaginary part √(-D)/(2a).

For the last example, the roots are -2 + 2i and -2 - 2i. Complex answers are not a calculator failure. They say that no real number makes the polynomial equal zero. NIST's Digital Library of Mathematical Functions section on polynomial zeros gives broader reference material on roots and polynomial methods.

Vertex, symmetry, and word problems

For y = ax² + bx + c, the vertex is (h, k), where h = -b/(2a) and k is found by substituting h into the equation. If a is positive, the vertex is the minimum; if a is negative, it is the maximum. The line x = h splits the parabola into two matching halves. These features matter in applications because a root can represent a time, distance, or break-even point, while the vertex often represents the best or worst value.

Suppose a toy rocket's height in feet is h(t) = -16t² + 64t + 5. The launch height is 5 feet, and the coefficient -16 makes the path open downward. The landing time solves -16t² + 64t + 5 = 0. The positive root is about 4.08 seconds; the negative root describes the mathematical extension before launch and is not a physical time. The vertex occurs at t = 2 seconds and height 69 feet. NASA's ballistic flight equations explain the ideal assumptions behind projectile models.

For a business example, profit P(x) = -2x² + 80x - 300 has break-even roots when P is zero. Its vertex is at x = 20 and P = 500, so the model's highest profit is 500 currency units at 20 items, before checking whether the model is realistic over that sales range.

The calculator also runs the problem backwards. Give it a vertex and one point on the curve: with vertex (2, -3) and the point (0, 2), the opening value p works out to 1, so the equation is y = (x - 2)^2 - 3 = x^2 - 4x + 1, whose y-intercept is 1 and whose discriminant is 12.

Checks and common mistakes

First copy every term into standard form. An equation such as 3x² = 7x - 2 becomes 3x² - 7x + 2 = 0, so b is -7, not 7. Second, keep the entire b in b²: if b = -4, then b² = 16. Third, use 2a as the whole denominator. Dividing by a before applying the plus and minus branches changes both roots.

• Do not use the quadratic formula when a is zero without switching to the linear equation.
• Do not discard a negative root before checking what the variable means in the word problem.
• Do not round the discriminant before taking its square root.
• Use the vertex formula with the original a and b, including their signs.
• Substitute each displayed root back into ax² + bx + c and inspect the residual.

The calculator displays the discriminant, root sum, product, vertex, factor form when real, and a substitution check. Those independent views make a sign error easier to spot than a single answer would.

A compact reference for each coefficient

These formulas describe one equation from several angles. For 6x² + x - 2 = 0, D = 49, the roots are 1/2 and -2/3, their sum is -1/6, and their product is -1/3. The decimals shown by the widget are presentation values, not a replacement for exact fractions when a homework or proof requires them.

All calculations run in your browser, and nothing you type is uploaded or stored. For additional algebra practice, the site's exponent calculator, average calculator, and percentage calculator cover nearby operations.