Quadratic Equation Solver

Solve $a x^{2} + b x + c = 0$ by factoring, completing the square or the formula — every step shown.

Try an example:

What a quadratic equation is

A quadratic equation has $x^{2}$ as its highest power and can be written as $a x^{2} + b x + c = 0$ with $a \neq = 0$ . It has at most two solutions, called roots: the $x$ -values where the parabola $y = a x^{2} + b x + c$ meets the $x$ -axis.

How to use the quadratic equation solver

Type or paste your problem into the box above — for example x^2 - 5x + 6 = 0. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

The quadratic formula

Every quadratic equation can be solved with the formula

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

The expression under the root, $b^{2} - 4 a c$ , is the discriminant. If it's positive there are two real roots; if it's zero there is one repeated root; if it's negative the roots are a pair of complex numbers.

Choosing a method

Rearrange so one side is zero, and read off $a$ , $b$ and $c$ .
Try factoring first: look for two numbers that multiply to $a c$ and add to $b$ . It's the fastest method when it works.
If the numbers aren't tidy, use the formula or complete the square, then check each root in the original equation.

Worked example: $2 x^{2} - 3 x - 5 = 0$

Identify the coefficients $a = 2$ , $b = - 3$ , $c = - 5$ .
Find the discriminant $b^{2} - 4 a c = 9 + 40 = 49$ . It's positive, so there are two real roots.
Apply the formula $x = \frac{3 \pm 49}{4} = \frac{3 \pm 7}{4}$
Simplify both roots $x = \frac{10}{4} = \frac{5}{2}$ or $x = \frac{- 4}{4} = - 1$ .
Check $2 (- 1)^{2} - 3 (- 1) - 5 = 2 + 3 - 5 = 0$ , and $2 (\frac{5}{2})^{2} - 3 (\frac{5}{2}) - 5 = 12.5 - 7.5 - 5 = 0$ . Both roots work.

Common mistakes to avoid

Losing the sign of $b$

With $b = - 3$ , $- b$ is $+ 3$ and $b^{2}$ is $+ 9$ . Write the substitution in brackets: $(- 3)^{2}$ .

Dividing only part of the top

The whole of $- b \pm b^{2} - 4 a c$ is divided by $2 a$ , not just the square root.

Not setting the equation to zero

For $x^{2} + 3 x = 10$ , the constant is $c = - 10$ , not $10$ . Move everything to one side first.

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Questions students ask

What if the discriminant is negative?

Then there are no real roots — the parabola never touches the $x$ -axis. The solver gives the two complex roots instead: $x^{2} + 2 x + 5 = 0$ gives $x = - 1 \pm 2 i$ .

Does it show factoring or just the formula?

It uses whichever method suits the equation and shows it in full. Ask for a specific method — solve by completing the square — and it uses that one.

Can it solve equations that aren't in standard form?

Yes. Type it as it is, such as (x + 1)(x - 3) = 5, and the first steps rearrange it into $a x^{2} + b x + c = 0$ .

Can it find the vertex too?

Yes. Ask for the vertex, the axis of symmetry or a description of the parabola in the same chat.