What a quadratic equation is
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b and c are numbers and a is not zero. The x² term is what makes it quadratic: if a were 0 the equation would be linear (bx + c = 0) with a single answer. A quadratic has at most two solutions, called roots or zeros, and its graph y = ax² + bx + c is a U-shaped curve called a parabola. The roots are the x-values where the parabola crosses the x-axis.
Before using the calculator, move every term to one side so the other side is 0. For example, 2x² + 3x = 2 becomes 2x² + 3x − 2 = 0, so a = 2, b = 3 and c = −2 (the default values).
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The ± means there are two answers: one with + and one with −. The expression under the square root, b² − 4ac, is called the discriminant (often written D or Δ). It tells you what kind of roots to expect before you finish the calculation:
| Discriminant | Roots | Graph |
|---|---|---|
| Positive (D > 0) | Two different real roots (rational if D is a perfect square and a, b, c are integers) | Crosses the x-axis twice |
| Zero (D = 0) | One repeated real root, x = −b ÷ 2a | Touches the x-axis at its vertex |
| Negative (D < 0) | Two complex conjugate roots, p ± qi | Never touches the x-axis |
Worked examples
1. Two rational roots: 2x² + 3x − 2 = 0
D = 3² − 4 × 2 × (−2) = 9 + 16 = 25, a perfect square, so the roots are rational. x = (−3 ± 5) ÷ 4, giving x = 0.5 and x = −2. The factored form is (x + 2)(2x − 1) = 0. Check: 2 × 0.5² + 3 × 0.5 − 2 = 0.5 + 1.5 − 2 = 0.
2. A repeated root: x² − 6x + 9 = 0
D = 36 − 36 = 0, so there is one root: x = 6 ÷ 2 = 3. The left side is a perfect square, (x − 3)², and the parabola just touches the x-axis at its vertex (3, 0).
3. Complex roots: x² + 2x + 5 = 0
D = 4 − 20 = −16. The square root of −16 is 4i, so x = (−2 ± 4i) ÷ 2 = −1 ± 2i. There are no real solutions; the parabola stays above the x-axis with its lowest point at (−1, 4).
4. Irrational roots: 2x² − 4x − 5 = 0
D = 16 + 40 = 56, which is not a perfect square. x = (4 ± √56) ÷ 4 = 1 ± √14 ÷ 2, so x ≈ 2.870829 and x ≈ −0.870829. These cannot be written as a product of factors with whole numbers, which is why the formula is needed.
5. A real-world quadratic: when does the ball land?
A ball thrown upward at 20 m/s from 1.5 m above the ground has height h = −4.9t² + 20t + 1.5 metres after t seconds (4.9 is half of the gravitational acceleration, 9.8 m/s²). It lands when h = 0, so a = −4.9, b = 20, c = 1.5. The roots are t ≈ 4.155 and t ≈ −0.074; time cannot be negative, so the ball lands after about 4.16 seconds. The vertex at t ≈ 2.04 s gives the maximum height, about 21.91 m.
Vertex, axis of symmetry and intercepts
The parabola's turning point, the vertex, lies halfway between the roots at x = −b ÷ 2a. Substitute that value back in to get its height: y = c − b² ÷ 4a. The vertical line through the vertex is the axis of symmetry. If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum, as in the ball example. The y-intercept is always c, the value of y when x = 0.
Completing the square turns the equation into vertex form, y = a(x − h)² + k, where (h, k) is the vertex. For 2x² + 3x − 2 that is 2(x + 0.75)² − 3.125, so the vertex is (−0.75, −3.125).
Other ways to solve a quadratic
- Factoring is fastest when the roots are whole numbers or simple fractions: x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or 3. Look for two numbers that multiply to c and add to b (when a = 1).
- Taking square roots works when there is no x term: x² − 9 = 0 gives x = ±3.
- Factoring out x works when c = 0: 3x² + 6x = 3x(x + 2), so x = 0 or −2.
- Completing the square always works and is how the quadratic formula itself is derived.
A quick check for any answer uses Vieta's formulas: the two roots add up to −b ÷ a and multiply to c ÷ a. In example 1, 0.5 + (−2) = −1.5 = −3 ÷ 2 and 0.5 × (−2) = −1 = −2 ÷ 2. The calculator shows both checks.
Common mistakes
- Not moving everything to one side first. x² = 5x − 6 must become x² − 5x + 6 = 0 before you read off a, b and c.
- Losing the sign of b or c. In 2x² − 4x − 5, b is −4 and c is −5. Then −b is +4, and b² is +16, never −16.
- Dividing only part of the top by 2a. The whole expression −b ± √D is divided by 2a, not just the square root.
- Rejecting complex roots as "no solution". A negative discriminant means no real solution; there are still two complex ones.
- Keeping answers that make no sense in context, such as a negative time or length. Both roots solve the equation, but only one may fit the problem.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves every equation of the form ax² + bx + c = 0 with a ≠ 0.
What does the discriminant tell you?
The discriminant b² − 4ac decides the type of roots: positive means two real roots, zero means one repeated real root, negative means two complex roots.
How do I solve 2x² + 3x − 2 = 0?
a = 2, b = 3, c = −2. The discriminant is 9 + 16 = 25, so x = (−3 ± 5) ÷ 4, which gives x = 0.5 and x = −2. In factored form: (x + 2)(2x − 1) = 0.
What if a = 0?
Then the equation is linear, bx + c = 0, with the single solution x = −c ÷ b. The calculator tells you this instead of using the quadratic formula.
How do I find the vertex of a parabola?
The x-coordinate is −b ÷ 2a and the y-coordinate is c − b² ÷ 4a. For 2x² + 3x − 2 the vertex is (−0.75, −3.125).
Sources
- OpenStax, College Algebra 2e (2021), section 2.5 Quadratic Equations and section 5.1 Quadratic Functions.
- OpenStax, Intermediate Algebra 2e (2020), chapter 9 Quadratic Equations and Functions.
- OpenStax, University Physics Volume 1 (2016), section 3.5 Free Fall (g ≈ 9.8 m/s²).
Last reviewed 19 October 2026. Results are estimates; see our disclaimer.