What is the quadratic formula?

The quadratic formula solves any equation of the form ax^2 + bx + c = 0 (with a 0). The roots are x = {-b {b^2 - 4ac}}{2a}. The produces the two solutions, and the expression b^2 - 4ac under the root is the discriminant.

What is the discriminant and why does it matter?

The discriminant is = b^2 - 4ac — the part under the square root. Its sign tells you the nature of the roots without solving: > 0 gives two distinct real roots, = 0 gives one repeated (double) root, and < 0 gives two complex conjugate roots.

Why does this calculator show a radical like 4 + √11 instead of a decimal?

When the discriminant is not a perfect square, the roots are irrational — their decimals never end or repeat. The exact answer is a simplified radical, for example 4 + {11}, and the decimal 7.31662 is only an approximation. The radical form is exact.

What happens when a = 0?

If a = 0 the x^2 term disappears and the equation is linear (bx + c = 0), not quadratic — the quadratic formula divides by 2a, so it is undefined. This calculator reports that case as "not quadratic"; enter a nonzero a.

How do complex roots work?

When < 0 the square root of a negative number is imaginary (i = {-1}), so the two roots are complex conjugates p qi. For example 5x^2 + 20x + 32 = 0 has roots -2 {2{15}}{5}i.

Is the quadratic formula the same as completing the square?

Yes — the quadratic formula is derived by completing the square on the general equation ax^2 + bx + c = 0. Completing the square rewrites it as a(x - h)^2 + k = 0; solving that for x gives exactly the quadratic formula.

Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula. Get exact roots as simplified radicals or complex numbers, plus the discriminant.

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The quadratic formula

A quadratic equation has the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are numbers and $a \neq 0$ . Its solutions — the values of $x$ that make it true — are given by the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The $\pm$ sign is what produces the two roots: one using $+$ , one using $-$ . Everything hinges on the quantity under the square root.

The discriminant decides everything

The expression under the radical, $\Delta = b^2 - 4ac$ , is called the discriminant. Its sign alone tells you how many roots there are and what kind — before you finish solving:

Discriminant	Roots	Example
$\Delta > 0$	Two distinct real roots	$x^2 - 8x + 5 = 0$
$\Delta = 0$	One repeated (double) real root	$x^2 - 4x + 4 = 0$
$\Delta < 0$	Two complex conjugate roots	$5x^2 + 20x + 32 = 0$

Tip: The discriminant is a shortcut: you can classify the roots of a quadratic just from

b^2 - 4ac

, without computing the roots themselves.

Worked example — two real roots

Solve $x^2 - 8x + 5 = 0$ , so $a = 1$ , $b = -8$ , $c = 5$ .

Discriminant: $\Delta = (-8)^2 - 4(1)(5) = 64 - 20 = 44$ . Since $44 > 0$ , there are two distinct real roots.
Substitute: $x = \dfrac{-(-8) \pm \sqrt{44}}{2(1)} = \dfrac{8 \pm \sqrt{44}}{2}$ .
Simplify the radical: $\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$ , so $x = \dfrac{8 \pm 2\sqrt{11}}{2} = 4 \pm \sqrt{11}$ .
Roots: $x_1 = 4 + \sqrt{11} \approx 7.31662$ and $x_2 = 4 - \sqrt{11} \approx 0.683375$ .

The two roots are exact as radicals; the decimals are rounded approximations.

Worked example — complex roots

Solve $5x^2 + 20x + 32 = 0$ , so $a = 5$ , $b = 20$ , $c = 32$ .

Discriminant: $\Delta = 20^2 - 4(5)(32) = 400 - 640 = -240$ . Since $-240 < 0$ , the roots are complex.
Substitute: $x = \dfrac{-20 \pm \sqrt{-240}}{2(5)} = \dfrac{-20 \pm \sqrt{-240}}{10}$ .
Handle the negative root: $\sqrt{-240} = \sqrt{240}\,i = 4\sqrt{15}\,i$ , since $240 = 16 \times 15$ . So $x = \dfrac{-20 \pm 4\sqrt{15}\,i}{10}$ .
Reduce: dividing through by 2, $x = -2 \pm \dfrac{2\sqrt{15}}{5}i \approx -2 \pm 1.54919i$ .

The two roots are complex conjugates — the same real part with opposite imaginary parts.

Simplifying the radical

The reason the answers show radicals like $2\sqrt{11}$ rather than long decimals is that a simplified radical is exact. To simplify $\sqrt{44}$ : factor out the largest perfect square, $44 = 4 \times 11$ , then pull out $\sqrt{4} = 2$ , leaving $2\sqrt{11}$ . The same step turns $\sqrt{240}$ into $4\sqrt{15}$ . See the Simplify Radicals calculator for that step on its own.

The connection to completing the square

The quadratic formula is not magic — it is what you get by completing the square on the general equation $ax^2 + bx + c = 0$ . Rewriting the left side as $a(x - h)^2 + k$ and solving for $x$ reproduces the formula exactly. If you want to see that intermediate vertex form, use the Completing the Square calculator; it is the same math, shown a different way.

Where quadratics show up

Quadratic equations model projectile motion (height over time), area and optimization problems, parabolic shapes in physics and engineering, and break-even points in finance. Any time a relationship squares one quantity, a quadratic — and this formula — is close behind.

Frequently asked questions

What is the quadratic formula?: The quadratic formula solves any equation of the form $ax^2 + bx + c = 0$ (with $a \neq 0$ ). The roots are $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The $\pm$ produces the two solutions, and the expression $b^2 - 4ac$ under the root is the discriminant.
What is the discriminant and why does it matter?: The discriminant is $\Delta = b^2 - 4ac$ — the part under the square root. Its sign tells you the nature of the roots without solving: $\Delta > 0$ gives two distinct real roots, $\Delta = 0$ gives one repeated (double) root, and $\Delta < 0$ gives two complex conjugate roots.
Why does this calculator show a radical like 4 + √11 instead of a decimal?: When the discriminant is not a perfect square, the roots are irrational — their decimals never end or repeat. The exact answer is a simplified radical, for example $4 + \sqrt{11}$ , and the decimal $7.31662$ is only an approximation. The radical form is exact.
What happens when a = 0?: If $a = 0$ the $x^2$ term disappears and the equation is linear ( $bx + c = 0$ ), not quadratic — the quadratic formula divides by $2a$ , so it is undefined. This calculator reports that case as "not quadratic"; enter a nonzero $a$ .
How do complex roots work?: When $\Delta < 0$ the square root of a negative number is imaginary ( $i = \sqrt{-1}$ ), so the two roots are complex conjugates $p \pm qi$ . For example $5x^2 + 20x + 32 = 0$ has roots $-2 \pm \frac{2\sqrt{15}}{5}i$ .
Is the quadratic formula the same as completing the square?: Yes — the quadratic formula is derived by completing the square on the general equation $ax^2 + bx + c = 0$ . Completing the square rewrites it as $a(x - h)^2 + k = 0$ ; solving that for $x$ gives exactly the quadratic formula.