How to Use the Quadratic Formula
Quadratic equations appear in physics, engineering, economics, architecture and many other fields. They describe the path of a thrown ball, the shape of a suspension bridge cable, the calculation of a projectile's range and much more.
The quadratic formula is the universal tool for solving these equations. Once you understand the structure, you can solve any quadratic equation with it in a few steps.
What Is a Quadratic Equation?
A quadratic equation is any equation of the form: ax² + bx + c = 0
Where a is the coefficient of x² (must not be zero), b is the coefficient of x, and c is the constant term.
Example: 2x² + 5x − 3 = 0. Here a = 2, b = 5, c = −3.
The graph of any quadratic equation is a parabola — a U-shaped or inverted U-shaped curve.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
The formula gives two solutions, one using the plus sign and one using the minus sign. These two solutions are the x-values where the parabola crosses the horizontal axis.
Step-by-Step: How to Use the Formula
- Write the equation in standard form. Make sure one side equals zero.
- Identify a, b and c. Write them down separately before plugging them in.
- Calculate the discriminant: b² − 4ac. This tells you how many solutions exist.
- Take the square root of the discriminant.
- Substitute everything into the formula and calculate both solutions.
- Check your answers by substituting them back into the original equation.
Worked Example 1: Two Real Solutions
Solve: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Discriminant: (−5)² − (4 × 1 × 6) = 25 − 24 = 1. Square root = 1.
x = (5 + 1) / 2 = 3 and x = (5 − 1) / 2 = 2
Solutions: x = 3 and x = 2. Check: 9 − 15 + 6 = 0. Correct.
Worked Example 2: One Repeated Solution
Solve: x² − 6x + 9 = 0
a = 1, b = −6, c = 9
Discriminant: 36 − 36 = 0. Square root = 0.
Solution: x = 3 (a repeated root).
Worked Example 3: No Real Solutions
Solve: x² + 2x + 5 = 0
a = 1, b = 2, c = 5
Discriminant: 4 − 20 = −16. A negative discriminant means the square root does not exist as a real number.
No real solutions. The parabola does not cross the horizontal axis at all.
The Discriminant: Your First Check
| Discriminant value | What it means |
|---|---|
| Greater than zero | Two distinct real solutions. The parabola crosses the axis at two points. |
| Equal to zero | One repeated real solution. The parabola just touches the axis. |
| Less than zero | No real solutions. The parabola does not touch the axis. |
When to Use the Quadratic Formula vs Other Methods
| Method | When to use it |
|---|---|
| Factoring | Works quickly when the equation factors neatly. Not always possible. |
| Completing the square | A useful technique but more steps than the formula. |
| Quadratic formula | Always works for any quadratic equation. Best default method. |
| Graphing | Useful for visualising but not precise for irrational roots. |
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Frequently Asked Questions
What does it mean if the discriminant is negative?
A negative discriminant means the equation has no real solutions. The square root of a negative number is not a real number. The parabola sits entirely above or below the horizontal axis and never crosses it. In advanced maths, these solutions are called complex or imaginary numbers.
Do I always need to use the quadratic formula?
No. If the equation factors easily — like x² − 5x + 6 = (x − 2)(x − 3) — factoring is faster. But when the numbers do not factor neatly, the quadratic formula always works and is the safest default.
What are a, b and c and how do I identify them?
Rewrite the equation as ax² + bx + c = 0. The coefficient of x² is a. The coefficient of x is b. The standalone number is c. If there is no x term, b = 0. If there is no constant term, c = 0.
Can the quadratic formula give irrational answers?
Yes. If the discriminant is not a perfect square, the square root will be irrational. The answers will be in the form (−b ± √d) / 2a where d is not a perfect square. These are valid exact answers and can be left in this form or approximated as decimals.