Key takeaways
- The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a and solves any ax² + bx + c = 0.
- The discriminant b² − 4ac tells you how many real solutions there are before you finish.
- Put the equation in the form ax² + bx + c = 0 first, with 0 on one side.
- Wrap negative numbers in brackets when you square them: (−5)² = 25.
In this guide
What is a quadratic equation?
A quadratic equation is an equation where the highest power of x is 2. Its standard form is ax² + bx + c = 0, where a, b and c are numbers and a is not zero.
When you draw y = ax² + bx + c on a graph, you get a U-shaped curve called a parabola. The solutions of the equation, also called roots, are the points where the curve crosses the x-axis.
Why does the quadratic formula matter?
Some quadratics factor neatly, but many do not. The quadratic formula always works, whatever the numbers are, even when the answers are messy decimals.
Quadratics appear whenever something speeds up, slows down or curves: a thrown ball, the shape of a bridge arch, the area of a garden with a fixed fence, or the profit from pricing a product.
One formula, any quadratic. That is why it is one of the most famous formulas in school maths.
The formula
| Symbol | What it means | Example in 2x² + 3x − 2 = 0 |
|---|---|---|
| a | Number in front of x² (not 0) | 2 |
| b | Number in front of x | 3 |
| c | Plain number (constant) | −2 |
| ± | Do it twice: once with +, once with − | gives two roots |
| b² − 4ac | The discriminant | 25 |
The discriminant
The part under the square root, D = b² − 4ac, is called the discriminant. It tells you what kind of answers to expect:
| Discriminant | Number of real roots | What the graph does |
|---|---|---|
| D > 0 | Two different roots | Crosses the x-axis twice |
| D = 0 | One repeated root | Just touches the x-axis |
| D < 0 | No real roots (two complex roots) | Never reaches the x-axis |
How to use the quadratic formula step by step
- 1Rearrange to standard form. Get 0 on one side:
ax² + bx + c = 0. - 2Write down a, b and c with their signs.
- 3Find the discriminant
b² − 4ac. - 4Take the square root of the discriminant (if it is 0 or positive).
- 5Work out both roots: once with +, once with −, then divide by
2a. - 6Check by putting a root back into the equation.
Now try it yourself. Enter a, b and c in the calculator below to see the discriminant and both roots worked out.
Try it yourself
Pre-filled with the example — change any value.
The calculator lists both roots in its steps and gives the larger root as the main result. If the discriminant is negative, it shows the complex roots in the form p ± qi instead.
Common mistakes to avoid
- Not setting the equation to 0. For
x² + 2x = 8, first writex² + 2x − 8 = 0, so c = −8. - Dividing only the square root by 2a. The whole top,
−b ± √D, is divided by2a. - Using `4ac` with the wrong sign. If a or c is negative,
−4acbecomes positive. - Forgetting the second root. The ± gives two answers unless D = 0.
Tips and tricks
- Complex roots: for
x² + 2x + 5 = 0, D =4 − 20 = −16, so the roots are−1 ± 2i, where i is the square root of −1. - Repeated root:
x² − 6x + 9 = 0has D = 0 and the single root x = 3. - Quick check: the two roots always add up to
−b ÷ aand multiply toc ÷ a. For the example,0.5 + (−2) = −1.5 = −3 ÷ 2. - Related tools: see the linear equation solver when a = 0, or the simultaneous equations calculator for two unknowns.
Where you'll use it in real life
- Sport and physics: a ball thrown up at 20 m/s has height about
h = 20t − 5t². Setting h = 15 gives5t² − 20t + 15 = 0, so t = 1 s (going up) or t = 3 s (coming down). The projectile range calculator covers how far it travels. - Design: finding the size of a garden or frame with a given area.
- Business: finding the prices where profit is exactly zero.
- Graphs: finding where a parabola crosses the x-axis, alongside the slope calculator for straight lines.
Quick summary
Put the equation in the form ax² + bx + c = 0, read off a, b and c with their signs, and use x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant tells you whether you get two, one or no real roots. Use the calculator above to check each step of your working.
Frequently asked questions
When should I use the quadratic formula?
Use it for any equation of the form ax² + bx + c = 0, especially when it does not factor easily. It always works.
What if the discriminant is negative?
A negative discriminant means there are no real solutions. The two solutions are complex numbers that involve i, the square root of −1.
What if a is 0?
Then there is no x² term, so the equation is linear, not quadratic. Solve bx + c = 0 to get x = −c ÷ b.
How do I know which root is the right one?
Both roots solve the equation. In a real problem, pick the one that makes sense, for example a positive time or a positive length.
