How do I use the quadratic formula?

Step by step

1. Step 1: Put it in standard form

   Move everything to one side so it reads ax² + bx + c = 0.

2. Step 2: Identify a, b and c

   Include the signs. In 2x² + 4x − 6 = 0, a = 2, b = 4 and c = −6.

3. Step 3: Find the discriminant

   b² − 4ac = 16 + 48 = 64.

4. Step 4: Finish the formula

   x = (−4 ± √64) ÷ 4 = (−4 ± 8) ÷ 4, so x = 1 or x = −3.

What does the discriminant tell you?

The discriminant is the part under the square root, b² − 4ac. It tells you what kind of answers to expect before you finish the problem.

• Positive: two real roots. 2x² + 3x − 2 = 0 has a discriminant of 25, so x = 0.5 or x = −2.
• Zero: one repeated root. x² − 6x + 9 = 0 has a discriminant of 0, so x = 3.
• Negative: two complex roots. x² + 2x + 5 = 0 has a discriminant of −16, so x = −1 ± 2i.

What if the square root isn't a whole number?

Leave it in exact form or round it, whichever your question asks for. For x² + 4x + 1 = 0, the discriminant is 12, so x = −2 ± √3, which is about −0.268 and −3.732.

To check your work, enter the coefficients into this quadratic formula calculator. It returns both roots, real or complex, along with the discriminant, the vertex of the parabola and the step-by-step working.

What mistakes should I watch for?

• Dropping a negative sign: if b = −5, then −b = 5, and b² = 25.
• Dividing only part of the top: the 2a goes under the whole numerator, including −b.
• Skipping standard form: x² = 5x − 6 must become x² − 5x + 6 = 0 first.
• Using it when a = 0: then there's no x² term, and the equation is linear.

When should I factor instead?

Factor when the numbers are small and it's quick to spot: x² − 5x + 6 = (x − 2)(x − 3). The formula works on every quadratic, so use it when factoring isn't obvious. A perfect-square discriminant means the equation could have been factored over whole or rational numbers.