The quadratic formula solves equations in the form ax^2 + bx + c = 0 when a is not zero. Its structure also tells you how many real solutions to expect.
Identify the coefficients
Rewrite the equation in standard form and identify a, b, and c, including negative signs. The formula is x = (-b +/- sqrt(b^2 - 4ac)) / (2a).
Read the discriminant
The discriminant b^2 - 4ac is positive for two real solutions, zero for one repeated real solution, and negative for a pair of complex solutions.
Check the roots
Substitute each result back into the original equation. Rounding can make a near-zero check look slightly off, so retain extra digits during the calculation.
A worked example
Worked example: for x^2 - 5x + 6 = 0, a=1, b=-5, and c=6. The discriminant is 1, so the two roots are 2 and 3; substituting them back checks the result.
Normalize the equation before entering coefficients. Keep the negative sign attached to b and use extra precision before rounding the final roots.
The equation must be moved to one side before coefficients are identified. Checking the signs in the discriminant and substituting the roots into the original expression are two separate safeguards.
A graph can provide a useful visual check: the roots are the points where the parabola crosses or touches the horizontal axis.
The formula is reliable when the equation is normalized first and the discriminant is interpreted before rounding.
COMMON QUESTIONS
Frequently asked questions
What if a equals zero?
The equation is no longer quadratic and should be solved as a linear equation if b is nonzero.
Why are my roots complex?
A negative discriminant means there are no real roots under the entered coefficients, but complex roots still exist.