Math

The Quadratic Formula

2 min readMedium5-question drill

Some quadratic equations factor cleanly — and some don't. The quadratic formula is your guaranteed escape hatch: it solves EVERY quadratic, no factoring guesswork required.

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A quadratic equation is any equation you can write in the form ax² + bx + c = 0, where a, b, and c are numbers and a ≠ 0. The x² term is what makes it quadratic — that's the highest power.

Sometimes you can solve these by factoring (breaking the expression into two parentheses that multiply to zero). But factoring only works nicely when the numbers cooperate. When they don't, you reach for the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

This formula gives you the value(s) of x that make the equation true. Those values are called the solutions or roots.

Let's decode it:

a, b, c come straight from ax² + bx + c = 0. Match them up carefully — including their signs.
The ± symbol means you actually do the calculation twice: once with + and once with −. That's why a quadratic usually has two solutions.
The part under the square root, b² - 4ac, is called the discriminant. It's a quick health check:
- If b² - 4ac > 0 → two real solutions.
- If b² - 4ac = 0 → one real solution (the ± gives the same number twice).
- If b² - 4ac < 0 → no real solutions (you'd be square-rooting a negative).

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The solutions of a quadratic are where the parabola crosses the x-axis. Here y = x² - 4 crosses at x = -2 and x = 2 — two real solutions.

How to use it, step by step:

Rearrange the equation so one side equals 0.
Identify a, b, and c.
Plug them into the formula.
Simplify the discriminant first, then take the square root.
Compute both the + and − versions.

Two handy shortcuts from Vieta's formulas: for ax² + bx + c = 0, the sum of the solutions is -b/a and the product is c/a. So if a question asks for the sum of the roots and a = 1, the answer is just -b — no formula needed.

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Does the discriminant b² - 4ac equal a perfect square?

Yes ↓

→ Solutions are 'nice' — try factoring or the formula

No ↓

Is b² - 4ac negative?

Yes ↓

→ No real solutions

No ↓

→ Use the formula; answer has a square root

Use the discriminant to predict what your answers will look like before solving.

The formula always works, but if you spot an easy factor first, factoring is faster. Keep both tools ready.

Quick check

What are the solutions to x² + -12x + 32 = 0?

x = -4 and x = -8

x = 4 and x = -8

x = 4 and x = 8

x = 12 and x = 32

Worked examples

Example 1

What are the solutions to x² - 12x + 32 = 0?

Example 2

If x² - 8x + 7 = 0, what is the sum of the solutions?

Example 3

What is the positive solution to 2x² - 4x - 3 = 0? Round to the nearest hundredth.

Common pitfalls

Mishandling the sign of b

The formula starts with -b. If b = -12, then -b = +12 — students often forget the double negative and use -12 instead. Always substitute the full value including its sign, in parentheses.

Forgetting to set the equation equal to zero

If the equation looks like x² + 3x = 10, you must move the 10 over first: x² + 3x - 10 = 0. Pulling a, b, c before zeroing out gives wrong coefficients.

Only computing one of the ± branches

The ± means two separate calculations. Skipping one loses a solution — and the test often asks for the other root or the sum, so you need both.

Dividing only part of the numerator by 2a

The entire -b ± √(b²-4ac) is divided by 2a, not just the square root. Keep the whole numerator over the denominator.

Key takeaways

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any equation written as ax² + bx + c = 0.
Always rearrange to = 0 first, then read off a, b, c with their signs.
The discriminant b² - 4ac tells you how many real solutions exist: positive = 2, zero = 1, negative = 0.
Sum of roots = -b/a; product of roots = c/a (Vieta's formulas) — use these when asked only for a sum or product.
Try factoring first for clean numbers; use the formula when factoring fails.

Try it yourself

5 practice questions on The Quadratic Formula, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

The Quadratic Formula

You are here

→Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables →Nonlinear Functions →Equivalent Expressions

Worked examples

Common pitfalls

Key takeaways

Further reading

Try it yourself