How to Use the Quadratic Formula: Solving ax² + bx + c = 0

Setting up the formula correctly

The formula only works once the equation is in standard form: ax² + bx + c = 0, with zero on the right. If you start with 2x² + 3x = 5, first subtract 5 to get 2x² + 3x − 5 = 0, giving a = 2, b = 3, c = −5. The most common mistake is misreading a sign, so write each coefficient with its sign attached before you substitute.

Watch the −b term especially. If b is already negative, then −b becomes positive: for x² − 5x + 6 = 0, b = −5, so −b = +5. A careful setup — a = 1, b = −5, c = 6 — turns the formula into x = (5 ± √(25 − 24)) / 2. Getting the signs right at this stage removes almost every error that follows.

What the discriminant tells you

The discriminant b² − 4ac decides the shape of the answer before you finish. If it is positive, the square root is a real number and the ± gives two distinct real roots. If it is exactly zero, the ± adds and subtracts nothing, so both roots coincide into a single repeated solution. If it is negative, the square root of a negative number is not real, so the equation has no real solutions — only complex ones.

Geometrically, the roots are where the parabola y = ax² + bx + c crosses the x-axis. Two real roots mean it crosses twice; a zero discriminant means it just touches the axis at its vertex; a negative discriminant means the parabola floats entirely above or below the axis and never crosses. Checking the sign of the discriminant first tells you what kind of answer to expect.

A full worked example

Solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5. The discriminant is 3² − 4 × 2 × (−5) = 9 + 40 = 49, which is positive, so expect two real roots. Its square root is 7, so x = (−3 ± 7) / (2 × 2) = (−3 ± 7) / 4. The plus branch gives (−3 + 7) / 4 = 1, and the minus branch gives (−3 − 7) / 4 = −2.5.

Always verify by substituting back. For x = 1: 2(1) + 3(1) − 5 = 2 + 3 − 5 = 0. For x = −2.5: 2(6.25) + 3(−2.5) − 5 = 12.5 − 7.5 − 5 = 0. Both check out. This substitution step catches sign slips and arithmetic mistakes, and it takes only a moment compared with re-solving from scratch.

Frequently asked questions

What if a equals zero?

Then it is not a quadratic. If a = 0 the x² term vanishes and you are left with the linear equation bx + c = 0, solved simply by x = −c / b. The quadratic formula would divide by 2a = 0, which is undefined, so use the linear method instead.

Can I always factor instead of using the formula?

Not always. Factoring is quick when the roots are simple whole numbers or neat fractions, but many quadratics have irrational or ugly roots that resist factoring. The formula works for every quadratic, so it is the reliable fallback when factoring does not click.

What does a repeated root mean?

It means the two solutions are equal because the discriminant is zero. For x² − 4x + 4 = 0 the discriminant is 16 − 16 = 0, so x = 2 is the only root, counted twice. The parabola touches the x-axis at exactly one point, its vertex.

How do I handle a negative discriminant?

There are no real solutions, so if your course only wants real answers, you state that. If complex numbers are allowed, write the square root of the negative discriminant using i, where i² = −1, and the ± then gives a pair of complex conjugate roots.