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Quadratic Formula Calculator with Steps

Enter a, b, and c and get both roots, the discriminant, and the vertex — with factoring, completing the square, and the formula compared on the same equation.

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How to use it

Type the equation, not just the coefficients — 2x^2-7x+3=0 is enough, and it does not need to be arranged with zero on the right first. Say if you want a particular method, otherwise the working picks the fastest one and explains the choice. Ask for the vertex or the intercepts in the same question and they come out of the same rearrangement. For quadratics that arrive inside a longer problem, use the algebra solver instead.

The method behind it

The quadratic formula is not a fact to memorise. It is what completing the square looks like when you do it once, in general, and keep the letters. Start from the general equation with $a \neq = 0$ .

a x^{2} + b x + c = 0

Divide through by $a$ so the square term is bare, and move the constant across.

x^{2} + \frac{b}{a} x = - \frac{c}{a}

Add the square of half the $x$ coefficient to both sides. That specific number is what turns the left side into a perfect square, which is the entire trick.

x^{2} + \frac{b}{a} x + (\frac{b}{2 a})^{2} = (\frac{b}{2 a})^{2} - \frac{c}{a}

The left side now folds up. Put the right side over the common denominator $4 a^{2}$ .

(x + \frac{b}{2 a})^{2} = \frac{b ^{2} - 4 a c}{4 a ^{2}}

Take square roots of both sides — this is where the plus-or-minus is born, because both signs square to the same thing — and subtract $b /2 a$ .

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Two things fall straight out of that derivation. The quantity $Δ = b^{2} - 4 a c$ under the radical decides everything: positive gives two distinct real roots, zero gives one repeated root, and negative gives a conjugate pair of complex roots. And $- b /2 a$ is the midpoint of the two roots, which is exactly the axis of symmetry of the parabola and the $x$ -coordinate of its vertex.

Two shortcuts are worth carrying. The roots always satisfy Vieta's relations, which give a five-second check on any answer.

x_{1} + x_{2} = - \frac{b}{a}, x_{1} x_{2} = \frac{c}{a}

A fuller treatment of when each method wins is in the quadratic formula: when to use it, and the factoring patterns themselves are laid out in factoring quadratics: the four patterns.

Worked examples

Worked example

2 x^{2} - 7 x + 3 = 0

1
Read off the coefficients. Keep the sign attached to b; that is where most errors start.
$a = 2, b = - 7, c = 3$
2
Compute the discriminant. It is 25, a perfect square, so the roots are rational and this equation also factors.
$Δ = (- 7)^{2} - 4 (2) (3) = 49 - 24 = 25$
3
Substitute into the formula. Note that −b is +7 because b was negative.
$x = \frac{7 \pm 25}{4} = \frac{7 \pm 5}{4}$
4
Split the plus-or-minus into the two cases.
$x = \frac{12}{4} = 3 or x = \frac{2}{4} = \frac{1}{2}$
5
Check with Vieta: the roots should add to 7/2 and multiply to 3/2. They do. The factored form is (x − 3)(2x − 1) = 0.
$3 + \frac{1}{2} = \frac{7}{2} = - \frac{b}{a}, 3 \cdot \frac{1}{2} = \frac{3}{2} = \frac{c}{a}$

Answer

x = 3 or x = \frac{1}{2}

Worked example

x^{2} + 4 x + 7 = 0

1
Coefficients first, then the discriminant. It is negative, so there are no real roots and no integer factorisation to look for.
$a = 1, b = 4, c = 7 \Rightarrow Δ = 16 - 28 = - 12$
2
Substitute anyway. The formula still works; the square root simply leaves the real numbers.
$x = \frac{- 4 \pm - 12}{2}$
3
Simplify the radical. Twelve is four times three, so the square root of −12 is 2i times the square root of 3.
$- 12 = 43 i = 2 i 3$
4
Divide every term of the numerator by 2, not just the first one.
$x = \frac{- 4 \pm 2 i 3}{2} = - 2 \pm i 3$
5
Verify the first root by substitution: the i² term contributes −3, and the real parts cancel exactly.
$(- 2 + i 3)^{2} + 4 (- 2 + i 3) + 7 = (1 - 4 i 3) - 8 + 4 i 3 + 7 = 0$

Answer

x = - 2 + i 3 or x = - 2 - i 3

Which method should you use?

The formula always works, which is precisely why people reach for it on equations that would take one line by other means. Compute the discriminant first and let it choose.

Factor, complete the square, or use the formula

Write the equation as ax² + bx + c = 0 first. Which coefficients are actually there?

The discriminant, on the graph

The roots of $a x^{2} + b x + c = 0$ are where the parabola $y = a x^{2} + b x + c$ crosses the horizontal axis, so the discriminant is really a statement about height. Start from $y = x^{2} - 2$ , which cuts the axis twice, and slide $c$ upward. At $c = 0$ the two crossings merge into one — that is $Δ = 0$ . Above that the curve lifts clear of the axis and the real roots are gone, though the complex pair is still there.

Slide the coefficients and watch the roots appear and vanish

y = 1x² + 0x + -2

Common mistakes

Reading coefficients before rearranging

From $x^{2} = 5 x + 6$ people take $b = 0$ and $c = 0$ . The formula only describes an equation set equal to zero. Move everything to one side first to get $x^{2} - 5 x - 6 = 0$ , so $b = - 5$ and $c = - 6$ .

Losing the sign in −b

When $b$ is negative, $- b$ is positive. With $b = - 7$ the numerator starts $+ 7$ , not $- 7$ . Writing the substitution with brackets around every coefficient, as in $- (- 7)$ , removes the guesswork.

Dividing only part of the numerator

$\frac{- 4 \pm 2 i 3}{2}$ is $- 2 \pm i 3$ , not $- 4 \pm i 3$ . The fraction bar runs under the whole numerator, so every term gets divided by $2 a$ .

Practice

Answers are checked here — nothing is sent anywhere.

1
$Solve x^{2} - 5 x + 6 = 0$
2
$Solve 3 x^{2} + 2 x - 1 = 0$
3
$Find the discriminant of 2 x^{2} + 3 x + 5$

Frequently asked questions

Will it show factoring as well as the formula?

Ask for it and yes. When the discriminant is a perfect square the equation factors over the integers, and seeing the same equation solved both ways is the fastest way to learn which one to reach for next time.

What happens when the discriminant is negative?

You get two complex conjugate roots, written in the form p ± qi. The parabola has no x-intercepts in that case, which is the graphical statement of the same fact.

Can the coefficients be fractions or decimals?

Yes. Multiplying through by the common denominator first usually keeps the arithmetic under the radical cleaner, and the working shows that step rather than pushing fractions through the whole calculation.

How do I check my roots without redoing the work?

Use Vieta's relations. The two roots must add to −b/a and multiply to c/a. Both checks take seconds and catch nearly every sign error in the formula.

What does it cost?

The first solution is free with no account needed. A free account includes three solutions a day, and Gauth Plus removes the cap for $11.99 a month with a three-day free trial.

Keep going

Algebra solverQuadratics inside larger problems: systems, inequalities, and rational equations.The quadratic formula: when to use itFour methods compared, and how the discriminant tells you which to pick.Factoring quadratics: the four patternsRecognise the pattern instead of guessing pairs of numbers.Factoring calculatorFactor the same expression completely, GCF and all, before solving.