Quadratic Formula Calculator
- Free
- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This quadratic formula calculator solves any equation of the form \( ax^2 + bx + c = 0 \). Enter the three coefficients and it returns the discriminant, both roots in exact form and as decimals, the factored form, the vertex, the axis of symmetry and the y-intercept. When the discriminant is negative it gives the complex roots \( p \pm qi \) instead of an error.
It is built for algebra, pre-calculus and calculus students checking homework, and for anyone who needs the zeros of a parabola quickly. It’s free and needs no sign-up.
How to use the quadratic formula calculator
- Rearrange your equation so one side is zero: \( ax^2 + bx + c = 0 \).
- Enter a, b and c. Include the signs: for \( 2x^2 - 7x + 3 = 0 \), enter \( a = 2 \), \( b = -7 \), \( c = 3 \). Fractions such as
1/2and decimals are accepted. - Press Solve. The first row is the discriminant \( \Delta = b^2 - 4ac \), which tells you how many real roots there are.
- Read the roots, the factored form (when the roots are rational) and the vertex. The vertex is the minimum point when \( a > 0 \) and the maximum point when \( a < 0 \).
Formula
The calculator uses the quadratic formula, which comes from completing the square on \( ax^2 + bx + c = 0 \):
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The expression under the root is the discriminant:
| Discriminant | Roots | Graph |
|---|---|---|
| \( \Delta > 0 \) | Two different real roots | Crosses the x-axis twice |
| \( \Delta = 0 \) | One repeated real root | Touches the x-axis at the vertex |
| \( \Delta < 0 \) | Two complex roots | Never touches the x-axis |
The vertex sits halfway between the roots, at
$$x_v = -\frac{b}{2a}, \qquad y_v = c - \frac{b^2}{4a}$$
Worked example: two rational roots
Solve \( 2x^2 - 7x + 3 = 0 \). These are the values loaded in the calculator above.
Discriminant. \( \Delta = (-7)^2 - 4(2)(3) = 49 - 24 = 25 \). It’s a perfect square, so the roots are rational.
Roots. \( x = \frac{7 \pm 5}{4} \), so \( x_1 = \frac12 \) and \( x_2 = 3 \).
Factored form. \( 2\left(x - \tfrac12\right)(x - 3) = (2x - 1)(x - 3) \).
Vertex. \( x_v = \frac{7}{4} \) and \( y_v = 3 - \frac{49}{8} = -\frac{25}{8} \). The parabola opens upward, so \( -\frac{25}{8} \) is its minimum value.
Worked example: complex roots
Solve \( x^2 + 2x + 5 = 0 \). The discriminant is \( 4 - 20 = -16 \), so there are no real roots. The formula still works with \( \sqrt{-16} = 4i \):
$$x = \frac{-2 \pm 4i}{2} = -1 \pm 2i$$
The vertex is \( (-1, 4) \). It lies above the x-axis and the parabola opens upward, which is why the graph never crosses it.
Worked example: irrational roots
For \( 2x^2 + 4x - 1 = 0 \), \( \Delta = 16 + 8 = 24 \). The calculator shows the exact form \( x = \frac{-4 \pm 2\sqrt6}{4} \), which simplifies to \( \frac{-2 \pm \sqrt6}{2} \), and the decimals \( x \approx 0.2247 \) and \( x \approx -2.2247 \).
Tips and common mistakes
- Dropping a sign. In \( 3x^2 - 5x - 2 = 0 \), \( b = -5 \), so \( -b = 5 \) and \( b^2 = 25 \), not \( -25 \).
- Dividing only part of the top by 2a. The whole numerator \( -b \pm \sqrt{\Delta} \) is divided by \( 2a \).
- Forgetting to set the equation to zero. \( x^2 + 3x = 10 \) must become \( x^2 + 3x - 10 = 0 \) before you read off \( c = -10 \).
- Using the formula when a = 0. Then the equation is linear, \( bx + c = 0 \), with the single root \( x = -c/b \).
- Check by substitution. Plugging \( x = 3 \) into \( 2x^2 - 7x + 3 \) gives \( 18 - 21 + 3 = 0 \).
Where quadratics show up in calculus
Setting a derivative equal to zero often produces a quadratic. For \( f(x) = 2x^3 - 3x^2 - 12x \), \( f'(x) = 6x^2 - 6x - 12 = 0 \) gives \( x = -1 \) and \( x = 2 \), the critical points (see how to find critical points). Projectile heights \( h(t) = -4.9t^2 + v_0t + h_0 \) are quadratics too, and the projectile motion calculator solves them for you.
Related guides
- Quadratic Formula: Derivation, Discriminant and Examples
- Polynomials: Degree, Roots and End Behavior
- How to Find Critical Points
- Domain and Range of a Function
Further reading
- Quadratic formula (Wikipedia) — the derivation by completing the square and the history of the formula.
- Solving Quadratic Equations, Part II (Paul’s Online Math Notes) — completing the square, the quadratic formula and the discriminant with many examples.
FAQ
What is the quadratic formula?
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It gives both solutions of \( ax^2 + bx + c = 0 \) for any \( a \neq 0 \).
How do I know how many solutions a quadratic has?
Look at the discriminant \( b^2 - 4ac \). Positive means two real solutions, zero means one repeated solution, and negative means two complex solutions.
Can the calculator give complex roots?
Yes. When \( b^2 - 4ac < 0 \), it shows the roots as \( p \pm qi \), for example \( -1 \pm 2i \) for \( x^2 + 2x + 5 = 0 \).
How do I find the vertex of a parabola?
Use \( x = -\frac{b}{2a} \), then substitute that x into the equation to get y. The calculator does both and also states whether the vertex is a minimum or a maximum.