Quadratic Equation Calculator

Q: What is the quadratic formula?

For ax^2 + bx + c = 0, the solutions are x = (-b ± √(b^2 - 4ac))/(2a). It works for every quadratic, including ones that do not factor.

Q: What does the discriminant tell me?

If b^2 - 4ac > 0 there are two different real roots, if it equals 0 there is one double root, and if it is negative there are two complex roots.

Q: Can the calculator give complex roots?

Yes. For a negative discriminant the roots are written as p ± qi; for x^2 + 2x + 5 = 0 they are -1 ± 2i.

Q: How do I find the vertex of the parabola?

The vertex is at x = -b/(2a); substitute that value to get the y-coordinate. The calculator shows both.

Q: Why can't a be zero in a quadratic equation?

If a = 0 the x^2 term vanishes and the equation becomes linear, bx + c = 0, with at most one solution. The quadratic formula would also divide by zero, so the calculator asks for a non-zero a.

About the Quadratic Equation Calculator

A quadratic equation has the form $ax^2 + bx + c = 0$ with $a \neq 0$ . Its graph is a parabola, and the solutions are the x-intercepts.

The discriminant $\Delta = b^2 - 4ac$ tells you what kind of roots to expect: two real roots if it is positive, one repeated root if it is zero, and two complex conjugate roots if it is negative. The calculator keeps radicals exact, so $2x^2 + 4x - 3 = 0$ gives $x = \frac{-2 \pm \sqrt{10}}{2}$ .

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two: the highest power of the unknown is $x^2$ . Written in standard form it is $ax^2 + bx + c = 0$ , where $a$ , $b$ and $c$ are numbers and $a$ is not zero. The coefficient $a$ controls how wide the parabola is and whether it opens upward or downward, $b$ shifts it sideways, and $c$ is where it crosses the y-axis.

Because the graph of $y = ax^2 + bx + c$ is a parabola, it can cross the x-axis twice, touch it once at its vertex, or miss it altogether. Those three pictures correspond to two real solutions, one repeated solution and no real solutions, and the discriminant decides which picture applies before any root is computed.

How to use the Quadratic Equation Calculator

Rewrite your equation so one side is 0.
Enter the coefficients a, b and c. Fractions and decimals are accepted.
Press Solve to see the discriminant, both roots, the factored form and the vertex.

Formulas

Quadratic formula

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

Discriminant

\Delta = b^2 - 4ac

Vertex

\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)

Vieta's formulas

x_1 + x_2 = -\frac{b}{a},\qquad x_1 x_2 = \frac{c}{a}

Worked example: x² − 3x − 10 = 0

The worked example solves $x^2 - 3x - 10 = 0$ , so $a = 1$ , $b = -3$ and $c = -10$ . The discriminant is $(-3)^2 - 4 \cdot 1 \cdot (-10) = 49$ , a positive perfect square, so there are two rational roots. The formula gives $\frac{3 \pm 7}{2}$ , that is $x = -2$ and $x = 5$ . The calculator also finds the vertex at $\left(\frac32, -\frac{49}{4}\right)$ and the factored form $(x + 2)(x - 5)$ , and confirms with Vieta's formulas that the roots add to 3 and multiply to −10.

The solution below is generated by the same engine that powers the calculator above.

Final answer

x_1 = -2,\quad x_2 = 5

How we read your problem

Solve a quadratic equation in standard form with the quadratic formula.

x^2 - 3x - 10 = 0

Step-by-step solution

Identify the coefficients of ax² + bx + c = 0
$x^2 - 3x - 10 = 0 \quad\Rightarrow\quad a = 1,\ b = -3,\ c = -10$
Compute the discriminant
$\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-10) = 49$
Δ > 0, so there are two distinct real roots
Apply the quadratic formula
$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{3 \pm \sqrt{49}}{2} = \frac{3 \pm 7}{2}$
Simplify each root
$x_1 = \frac{3 - 7}{2} = -2,\quad x_2 = \frac{3 + 7}{2} = 5$
Find the vertex of the parabola
The parabola opens upward, so the vertex is a minimum.
$h = -\frac{b}{2a} = \frac{3}{2},\quad k = c - \frac{b^2}{4a} = -\frac{49}{4}$
Factored form
$x^2 - 3x - 10 = (x + 2)(x - 5)$

Formulas used

Quadratic formula
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant
$\Delta = b^2 - 4ac$
Vertex
$\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)$

The discriminant is positive, so the parabola crosses the x-axis at two points.

Verification

Substitute x = -2: 1·(-2)² + (-3)·(-2) + (-10) = 0
Substitute x = 5: 1·5² + (-3)·5 + (-10) = 0
Vieta: sum of roots = 3 = −b/a = 3; product = -10 = c/a = -10

Detailed explanation

The three cases of the discriminant

When $\Delta > 0$ the square root is a real number and the $\pm$ produces two different roots. If $\Delta$ is also a perfect square, as with 49 in the example, the roots are rational and the quadratic factors over the integers; otherwise they contain a radical, as in $2x^2 + 4x - 3 = 0$ , whose roots are $\frac{-2 \pm \sqrt{10}}{2}$ . When $\Delta = 0$ the two roots coincide: $x^2 - 6x + 9 = 0$ has the double root $x = 3$ and the parabola just touches the axis. When $\Delta < 0$ there are no real roots, and the calculator gives the complex pair, for example $-1 \pm 2i$ for $x^2 + 2x + 5 = 0$ .

Where the formula comes from

The quadratic formula is what you get by completing the square on the general equation. Dividing by $a$ and moving $c$ gives $x^2 + \frac{b}{a}x = -\frac{c}{a}$ ; adding $\left(\frac{b}{2a}\right)^2$ to both sides makes the left a perfect square, $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$ . Taking square roots and subtracting $\frac{b}{2a}$ produces the formula. That derivation also explains the vertex: the parabola is symmetric about $x = -\frac{b}{2a}$ , the midpoint of the two roots.

Reading the vertex and the graph

The vertex is the turning point of the parabola. Its x-coordinate is $-\frac{b}{2a}$ , and substituting that into the equation gives the y-coordinate, the minimum value when $a > 0$ and the maximum when $a < 0$ . In the example the vertex is at $\left(\frac32, -\frac{49}{4}\right)$ , so the expression $x^2 - 3x - 10$ is never smaller than $-12.25$ . This is what makes quadratics useful in optimisation problems, such as finding the highest point of a thrown ball or the price that maximises revenue.

The vertex form $a(x - h)^2 + k$ shows the same information directly, with $(h, k)$ as the vertex.

Common mistakes

Using $-b^2$ instead of $b^2$ when $b$ is negative: $(-3)^2 = 9$ .
Dividing only $-b$ by $2a$ instead of the whole numerator.
Entering coefficients before moving every term to one side.
Forgetting the ± and reporting only one root.
Writing complex roots when only real solutions were asked for, or the reverse; read what the question wants.

Tips

If $b$ is even, the formula simplifies to $x = \frac{-b/2 \pm \sqrt{(b/2)^2 - ac}}{a}$ .
Check your roots with Vieta: their sum should be $-\frac{b}{a}$ .
If $\Delta$ is a perfect square, the quadratic factors over the integers.
Divide all three coefficients by a common factor first; the roots do not change and the arithmetic is lighter.
The vertex lies halfway between the two roots, which gives a quick check on both.

Frequently asked questions

What is the quadratic formula?

For $ax^2 + bx + c = 0$ , the solutions are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . It works for every quadratic, including ones that do not factor.

What does the discriminant tell me?

If $b^2 - 4ac > 0$ there are two different real roots, if it equals 0 there is one double root, and if it is negative there are two complex roots.

Can the calculator give complex roots?

Yes. For a negative discriminant the roots are written as $p \pm qi$ ; for $x^2 + 2x + 5 = 0$ they are $-1 \pm 2i$ .

What if a = 0?

Then the equation is linear, not quadratic. Use the equation solver or the solve-for-x calculator instead.

How do I find the vertex of the parabola?

The vertex is at $x = -\frac{b}{2a}$ ; substitute that value to get the y-coordinate. The calculator shows both.

Why keep roots as square roots instead of decimals?

Exact radicals keep full precision and show the structure of the answer. A decimal approximation is listed alongside for convenience.

What does a double root mean?

It means the discriminant is zero and both roots are the same number. The parabola touches the x-axis at its vertex without crossing it, as with $x^2 - 6x + 9 = 0$ , which has the double root $x = 3$ .

When should I factor instead of using the formula?

Factor when the discriminant is a perfect square and the numbers are small, since the roots are then rational. The formula always works, so use it when factoring is not obvious.

Why can't a be zero in a quadratic equation?

If $a = 0$ the $x^2$ term vanishes and the equation becomes linear, $bx + c = 0$ , with at most one solution. The quadratic formula would also divide by zero, so the calculator asks for a non-zero $a$ .

Solve your quadratic equation

Enter a, b and c above to get the discriminant, exact roots, the vertex and the factored form, each with its working shown.

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