The Quadratic Formula, Explained

Some quadratics factorise in a few seconds. Others just don't — the numbers don't cooperate, and guessing two factors that multiply and add to the right values stops being realistic. That's what the quadratic formula is for: it solves any quadratic equation, no factorising required.

For an equation in the form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / (2a)

The ± is doing real work here — it's why a quadratic can have two different answers, one from + and one from −.

Working through an example

Take x² + 2x − 8 = 0. Matching it to ax² + bx + c = 0 gives a = 1, b = 2, c = −8.

Step 1 — find the discriminant. This is the part under the square root, b² − 4ac:

b² − 4ac = (2)² − 4(1)(−8) = 4 + 32 = 36

Step 2 — take the square root. √36 = 6.

Step 3 — substitute into the formula:

x = (−2 ± 6) / 2

Step 4 — split into the two cases:

• x = (−2 + 6) / 2 = 2
• x = (−2 − 6) / 2 = −4

So the solutions are x = 2 and x = −4. You can check this by factorising the same equation as (x − 2)(x + 4) = 0 — same two answers, different route.

The discriminant tells you what's coming

That b² − 4ac piece is called the discriminant, and it's worth calculating before you go any further, because it tells you how many real solutions to expect:

• Positive (> 0) — two real solutions, like the example above.
• Zero (= 0) — exactly one real solution (the two ± cases land on the same number).
• Negative (< 0) — no real solutions, because you can't take the square root of a negative number.

Checking the discriminant first is a good habit — if it's negative, you know immediately there's nothing more to solve for.

When to reach for the formula

Factorising is usually faster when the numbers are small and cooperative. The quadratic formula is the reliable fallback when they're not — it works on every quadratic, including the ones with awkward or non-integer solutions that factorising would never find cleanly.

In short

• The quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), solves any equation of the form ax² + bx + c = 0.
• The discriminant b² − 4ac tells you how many real solutions exist before you finish the calculation: positive means two, zero means one, negative means none.
• Reach for it when an equation won't factorise easily.

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