Method · Algebra
How to Use the Quadratic Formula: Step by Step
Five steps, in order. Four of the five mistakes people make happen in the first two.
The five steps
Knowing how to use the quadratic formula is mostly a matter of doing these in order and not skipping the first.
- Rearrange into standard form: ax² + bx + c = 0, everything on one side.
- Read off a, b and c with their signs.
- Evaluate the discriminant b² − 4ac on its own.
- Substitute into the formula and simplify the root.
- Split the ± into two answers, and check them.
A worked example with rational roots
Start with 2x² + 7x + 3 = 0, where a = 2, b = 7 and c = 3.
- b² − 4ac = 49 − 24 = 25the discriminant is a perfect square
- x = (−7 ± 5) ÷ 4√25 = 5, and 2a = 4
- x = −2⁄4 = −½taking the plus sign
- x = −12⁄4 = −3taking the minus sign
Because the discriminant was a perfect square, both roots came out rational — which also means this one would have factored, as (2x + 1)(x + 3).
A worked example with irrational roots
Now x² − x − 1 = 0, where a = 1, b = −1 and c = −1. Note that −b is +1 here: this is the sign step that catches people.
- b² − 4ac = 1 − (−4) = 5careful: 4ac is negative, so it is added
- x = (1 ± √5) ÷ 2−b is +1 because b was negative
- x ≈ 1.618 or −0.618as decimals, rounded at the end only
√5 does not simplify, so (1 ± √5)/2 is the exact answer and the decimals are the approximation. The positive root here is the golden ratio.
When the discriminant is zero
For x² + 2x + 1 = 0 the discriminant is 4 − 4 = 0. The ± then adds and subtracts nothing, so both roots collapse into one.
- b² − 4ac = 4 − 4 = 0zero discriminant
- x = −2 ÷ 2 = −1the ± disappears
This is called a repeated root. The quadratic is a perfect square, (x + 1)², and the parabola touches the x-axis rather than crossing it.
Simplifying the surd properly
When the discriminant is not a perfect square, the answer is usually left in exact form, and that form should be fully simplified.
- (−4 ± √48) ÷ 4the raw substitution
- (−4 ± 4√3) ÷ 4√48 = √(16 · 3) = 4√3
- −1 ± √3every term divides by 4
The last step divides the whole numerator, both the −4 and the 4√3, by the denominator. Dividing only one of them is the error that hides here.
Reading the coefficients without slipping
Before any arithmetic, write a, b and c down separately with their signs attached. The coefficients are what the whole formula runs on, and misreading one is unrecoverable.
| Equation | a | b | c | The trap |
|---|---|---|---|---|
| x² + 5x = 6 | 1 | 5 | −6 | not in standard form yet |
| 3 − x² = 2x | −1 | −2 | 3 | a is negative, and the terms are out of order |
| x² − 7 = 0 | 1 | 0 | −7 | a missing term means that coefficient is zero |
| x² − x − 1 = 0 | 1 | −1 | −1 | a bare −x means b = −1, not −0 |
The last row is the negative b case worth dwelling on. When b = −1, the formula's −b becomes +1, and every later line depends on getting that one sign right.
Checking the two roots
There is a check faster than substituting back, and it is the one worth building into your habits. The sum of roots and the product of roots are both readable straight from the coefficients: they must add to −b/a and multiply to c/a.
| Equation | Roots should sum to | and multiply to |
|---|---|---|
| 2x² + 7x + 3 | −7/2 = −3.5 | 3/2 = 1.5 |
| x² − x − 1 | 1 | −1 |
For the first: −½ + −3 = −3.5 ✓ and −½ × −3 = 1.5 ✓. If either fails, the arithmetic went wrong and you know it in seconds rather than after re-solving the whole thing.
These two relationships are known as Vieta's formulas, and they hold for every quadratic. They are the fastest way to check your answer, and they also let you build a quadratic backwards from roots you already know: if the roots are 2 and 5, then b/a = −7 and c/a = 10, so x² − 7x + 10 = 0 is one equation with those roots.
Questions about how to use the quadratic formula
What do I do if the equation is not equal to zero?
Rearrange it first. Move every term to one side so the other side is zero, then read off a, b and c. The formula only applies to standard form.
What if b is negative?
Then −b is positive. For b = −3 the numerator starts with +3. This single sign is responsible for a large share of wrong answers.
Do I have to simplify the surd?
Usually yes. √48 should become 4√3, and if every term of the fraction then shares a factor, divide it out — all of the numerator, not part of it.
How can I check my roots quickly?
Add them and multiply them. The sum must equal −b/a and the product must equal c/a. Both take seconds and catch nearly every arithmetic slip.