The Complete Guide to Solving Quadratic Equations
By Arshi Khan, Founder of Just4Marketing · Last reviewed October 2026
Every quadratic equation ax² + bx + c = 0 can be solved with the same formula, but the discriminant inside that formula determines whether you get two real roots, one repeated root, or two complex roots. This guide walks through the formula, the discriminant, and the vertex.
What is the quadratic formula and why does it matter?
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are known numbers and a is never zero. The quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a, solves for x directly from those three coefficients, without needing to factor the equation by hand — which makes it the one method guaranteed to work on any quadratic, factorable or not.
The expression under the square root, b² − 4ac, is called the discriminant, and its sign alone tells you what kind of roots the equation has before you even finish the calculation.
What the discriminant tells you
| Discriminant (b² − 4ac) | Root type | Meaning |
|---|---|---|
| Positive | Two distinct real roots | The parabola crosses the x-axis at two points |
| Zero | One repeated real root | The parabola just touches the x-axis at its vertex |
| Negative | Two complex roots | The parabola never crosses the x-axis |
Things to double-check
Watch the sign of b when substituting
The formula uses −b, so if b is already negative, −b becomes positive. Dropping a negative sign here is the single most common manual error.
The ± means there are two answers, not one
Unless the discriminant is exactly zero, a quadratic equation always has two roots — one from +√ and one from −√.
The vertex x-coordinate is the average of the two real roots
When real roots exist, the axis of symmetry x = −b ÷ 2a always falls exactly halfway between them, which is a quick way to sanity-check your answer.
Complex roots always come in a conjugate pair
If one root is p + qi, the other is always p − qi — the real part stays the same and only the sign of the imaginary part flips.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves for x in any equation of the form ax² + bx + c = 0, as long as a is not zero.
What does a negative discriminant mean?
It means the equation has no real roots — the parabola never crosses the x-axis. Instead, the two roots are complex numbers of the form p ± qi, where i is the imaginary unit.
What is the vertex of a parabola and how is it found?
The vertex is the parabola's highest or lowest point, found at x = −b ÷ 2a (the axis of symmetry). Substituting that x back into the original equation gives the vertex's y-coordinate.
Can a ever be zero in a quadratic equation?
No — if a is zero, the x² term disappears and the equation becomes linear (bx + c = 0), which has at most one solution and is no longer a quadratic.