The quadratic formula, and what the discriminant tells you

Any equation of the form ax2 + bx + c = 0 is a quadratic, and one formula solves all of them. But there is a cheaper calculation to do first, which tells you how many answers exist before you go looking for them.

The formula

x = (−b ± √(b2 − 4ac)) ÷ 2a

The ± is why a quadratic usually has two solutions: you work it once with a plus and once with a minus. The expression under the square root, b2 − 4ac, is called the discriminant, and on its own it decides the shape of the answer.

Check the discriminant first

• Positive — two different real solutions. The curve crosses the horizontal axis twice.
• Zero — one repeated solution. The curve just touches the axis.
• Negative — no real solutions. The curve never reaches the axis, and you would be taking the square root of a negative number.

One subtraction saves you from hunting for roots that are not there.

A real example: x² − 3x + 2 = 0

Here a = 1, b = −3, c = 2.

1. Discriminant: (−3)2 − 4(1)(2) = 9 − 8 = 1. Positive, so two real solutions.
2. Square root of 1 is 1.
3. With the plus: (3 + 1) ÷ 2 = 2.
4. With the minus: (3 − 1) ÷ 2 = 1.

So x = 2 and x = 1. Check the first by substituting back: 22 − 3(2) + 2 = 4 − 6 + 2 = 0. Correct. Substituting back takes ten seconds and catches nearly every sign error, which is the most common mistake in the whole procedure — note that b was −3, so −b was +3.

A real example with a negative c: 2x² + 4x − 6 = 0

a = 2, b = 4, c = −6.

1. Discriminant: 42 − 4(2)(−6) = 16 + 48 = 64. Subtracting a negative added to it.
2. Square root of 64 is 8.
3. With the plus: (−4 + 8) ÷ 4 = 1.
4. With the minus: (−4 − 8) ÷ 4 = −3.

Checking x = −3: 2(9) + 4(−3) − 6 = 18 − 12 − 6 = 0. Correct.

The touching case: x² − 4x + 4 = 0

Discriminant: 16 − 16 = 0. The square root of zero is zero, so the plus and the minus give the same thing: 4 ÷ 2 = 2. One solution, x = 2, repeated. This is the quadratic that factors to (x − 2)2, and the discriminant told you so before you factored anything.

The no-real-answer case: x² + 2x + 5 = 0

Discriminant: 4 − 20 = −16. Negative, so there is no real number that satisfies this equation. Solutions do exist in the complex numbers, but if your x is a length, a price or a time, "no real solution" is the meaningful answer — and usually a sign that the situation you modelled cannot happen.

When a is zero

If a = 0 there is no x2 term and it is not a quadratic at all — it is linear, bx + c = 0, solved by x = −c ÷ b. Worth noticing because the formula itself breaks here: its denominator, 2a, would be zero. For 2x − 6 = 0 the answer is simply 6 ÷ 2 = 3.

Run your own numbers

The free Quadratic Solver takes a, b and c, reports both roots when they are real, and tells you the discriminant when they are not — entirely on your device.