How to use the Quadratic Equation
- Write your equation in the form ax² + bx + c = 0, moving everything to one side.
- Enter the coefficients a, b and c, including their signs.
- Read the discriminant first to see how many real roots to expect.
- Check the roots by substituting them back into the original equation.
How the calculation works
The quadratic formula comes from completing the square on the general form, which is why it works for every quadratic rather than only factorable ones. Its key part is the discriminant, b² − 4ac: positive gives two distinct real roots, zero gives one repeated root where the parabola touches the axis, and negative gives a conjugate pair of complex roots and a curve that never crosses the axis.
The vertex, at x = −b / 2a, is often the answer people actually want — it locates the maximum or minimum of the parabola, which is the optimal point in pricing and physics problems. If a is zero the expression is linear rather than quadratic and the formula does not apply, since division by 2a would be undefined.
x = (−b ± √(b² − 4ac)) / 2a ; discriminant Δ = b² − 4ac ; vertex x = −b / 2aSource: Derived by completing the square on the general quadratic; standard algebraic result.
Worked example
Solve 3x² − 11x + 6 = 0.
- Δ = (−11)² − 4(3)(6) = 121 − 72 = 49.
- √49 = 7, so the roots are rational.
- x = (11 + 7) / 6 = 3 and x = (11 − 7) / 6 = 0.6667.
Roots x = 3 and x = 2/3. Vertex at x = 11/6 ≈ 1.833, giving a minimum since a is positive.
Frequently asked questions
What does a negative discriminant mean?+
There are no real solutions — the parabola never crosses the x-axis. The two roots are complex conjugates, which matter in electrical engineering and signal analysis but represent no real intercept in a geometric problem.
Why can I not use the formula when a = 0?+
With a = 0 the equation is linear, bx + c = 0, and the formula would divide by zero. Solve it directly as x = −c / b.
Should I factor or use the formula?+
Factoring is faster when the roots are small integers. The formula always works, so it is the reliable choice for messy coefficients or when you need decimal answers.
How do I find the maximum or minimum value?+
Evaluate the expression at the vertex, x = −b / 2a. If a is positive that point is a minimum; if a is negative it is a maximum.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.