Quadratic formula
The quadratic formula solves any equation of the form a x squared plus b x plus c equals zero. It gives both roots directly from the three coefficients, working even when the equation does not factor neatly.
- x = the solutions, or roots, of the equation
- a, b, c = the coefficients of ax² + bx + c = 0
- b² − 4ac = the discriminant, which sets the number of real roots
What each part does
The piece negative b over 2a locates the axis of symmetry, the midpoint between the two roots. The square-root term measures how far the roots sit on either side of that midpoint. The plus-or-minus sign is what produces two solutions, one on each side.
The discriminant
The expression under the square root, b² − 4ac, is the discriminant. If it is positive there are two real roots; if it is zero there is one repeated root; if it is negative there are no real roots and the solutions are complex. Checking it first tells you what kind of answer to expect.
- Identify the coefficients: a = 1, b = 5, c = 6.
- Compute the discriminant: 5² − 4(1)(6) = 25 − 24 = 1.
- Take the square root: √1 = 1.
- Apply the formula: x = (−5 ± 1) / 2, giving x = −2 and x = −3.
Solve one instantly
Use the Quadratic Formula Calculator, or read How to Solve Quadratic Equations for all three methods.
FAQ
When do I use the quadratic formula?
For any equation of the form ax² + bx + c = 0, especially when it does not factor easily. It always produces the solutions.
What does the discriminant tell me?
The sign of b² − 4ac reveals whether there are two, one, or no real solutions before you finish solving.