The Quadratic Formula Explained, With a Projectile Example
October 21, 2026
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its solutions are given by x = (−b ± √(b² − 4ac)) ÷ 2a. The formula works for every quadratic, including those that cannot be factored neatly, and the expression under the square root, called the discriminant, tells you in advance how many real solutions exist.
Quadratics appear whenever a quantity depends on the square of another: projectile motion, areas, profit curves, and braking distances. This guide explains the formula, interprets the discriminant, and finishes with a physics problem.
Skip the math and get your answer instantly:
Open the Scientific Calculator →Using the Formula Step by Step
For x² − 5x + 6 = 0, a = 1, b = −5, c = 6. The discriminant is b² − 4ac = 25 − 24 = 1. Then x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, giving x = 3 and x = 2. Check: 3² − 15 + 6 = 0 and 2² − 10 + 6 = 0. Both work.
The same equation factors as (x − 2)(x − 3) = 0. Factoring is faster when it works, but the formula always works, even when the roots are irrational or the coefficients are awkward decimals.
What the Discriminant Tells You
If b² − 4ac is positive, there are two distinct real solutions. If it equals zero, there is one repeated real solution, the vertex touching the x-axis. If it is negative, there are no real solutions; the parabola never crosses the axis, and the solutions are complex numbers.
For example, x² + 2x + 5 = 0 has discriminant 4 − 20 = −16, so there are no real roots. The graph is a parabola that stays entirely above the x-axis.
| Discriminant | Number of real roots | Graph |
|---|---|---|
| Positive | 2 | Crosses the x-axis twice |
| Zero | 1 (repeated) | Touches the x-axis at the vertex |
| Negative | 0 | Does not reach the x-axis |
The Vertex and Axis of Symmetry
The parabola's turning point is at x = −b ÷ 2a, which lies midway between the two roots. Substituting back into the equation gives the maximum or minimum value. For y = x² − 5x + 6, the vertex is at x = 2.5 and y = 6.25 − 12.5 + 6 = −0.25. Since a is positive, it is a minimum.
In optimization problems, the vertex gives the best answer. For a profit equation with a negative leading coefficient, the vertex is the maximum profit.
Worked Example: A Ball Thrown Upward
A ball is thrown upward at 20 m/s from a height of 1.5 m. Ignoring air resistance, its height after t seconds is h = −4.9t² + 20t + 1.5 (using g = 9.8 m/s²). When does it hit the ground? Set h = 0: a = −4.9, b = 20, c = 1.5. The discriminant is 400 + 29.4 = 429.4, and √429.4 = 20.72.
t = (−20 ± 20.72) ÷ (−9.8). The positive solution is t = (−20 − 20.72) ÷ (−9.8) = 4.155 seconds. The negative root (−0.07 s) is not physical because it occurs before the throw. The peak is at t = −b ÷ 2a = 2.04 s, where h = −4.9(4.163) + 40.82 + 1.5 = 21.9 m.
Completing the Square and Where the Formula Comes From
The formula is derived by completing the square: divide by a, move the constant, add (b ÷ 2a)² to both sides, and take square roots. Seeing the derivation once helps you trust the formula and recognize forms such as vertex form, y = a(x − h)² + k, in which the vertex is (h, k).
Common Mistakes
Forgetting the sign of b when substituting (use −(−5) = 5 carefully), dividing only part of the numerator by 2a, dropping the ± and finding only one root, and forgetting to reject solutions that make no sense in context, such as negative times or lengths.
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a for the equation ax² + bx + c = 0.
What does a negative discriminant mean?
There are no real solutions; the parabola never meets the x-axis.
When should I factor instead?
When the numbers factor easily. Otherwise use the formula, which always works.
How do I find the vertex of a parabola?
x = −b ÷ 2a, then substitute to find y.
Why do I get two answers?
A parabola can cross the horizontal axis at two points. Choose the one that fits the situation.