Quadratic Formula Calculator
Solve ax² + bx + c = 0 instantly
The Quadratic Formula Calculator solves any quadratic equation of the form ax² + bx + c = 0. Enter the three coefficients a, b, and c, and it instantly finds both roots using the quadratic formula — showing each step clearly.
The quadratic formula is: x = (−b ± √(b² − 4ac)) / 2a
The key to understanding the result is the discriminant (D = b² − 4ac):
- D > 0 — Two distinct real roots. The parabola crosses the x-axis at two points.
- D = 0 — One repeated real root. The parabola just touches the x-axis at one point (vertex on x-axis).
- D < 0 — No real roots. The parabola doesn't cross the x-axis. The roots are complex numbers involving i (√−1).
Quadratic equations appear in physics (projectile motion), economics (profit maximisation), engineering, and geometry. Mastering this formula is one of the most useful skills in mathematics.
Example
Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6).
Step 1: Discriminant D = (−5)² − 4×1×6 = 25 − 24 = 1.
Step 2: Since D = 1 > 0, there are two distinct real roots.
Step 3: x = (5 ± √1) / 2 = (5 ± 1) / 2
x₁ = (5 + 1) / 2 = 3, x₂ = (5 − 1) / 2 = 2
Verification: (x−3)(x−2) = x² − 5x + 6 ✓
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a. It gives both roots (x₁ and x₂) of any quadratic equation.
What is the discriminant?
The discriminant is D = b² − 4ac. If D > 0, there are two real roots. If D = 0, there is one repeated root. If D < 0, there are two complex (imaginary) roots and no real solutions.
Can every quadratic equation be solved with the formula?
Yes. The quadratic formula works for any quadratic equation ax² + bx + c = 0 as long as a ≠ 0. It's more general than factoring, which only works for factorable equations.
What does it mean when a quadratic has complex roots?
Complex roots occur when the discriminant D < 0. The roots contain the imaginary unit i (√−1). This means the parabola y = ax² + bx + c never crosses the x-axis.
How do I solve x² − 3x − 10 = 0?
a=1, b=−3, c=−10. Discriminant D = (−3)² − 4(1)(−10) = 9 + 40 = 49. x = (3 ± √49) / 2 = (3 ± 7) / 2. So x₁ = (3+7)/2 = 5 and x₂ = (3−7)/2 = −2. Verification: (x−5)(x+2) = x² − 3x − 10 ✓.
What is the vertex of a parabola and how does it relate to the quadratic formula?
The vertex is the highest or lowest point of a parabola y = ax² + bx + c. Its x-coordinate is −b/(2a) — exactly the midpoint of the two roots from the quadratic formula. The y-coordinate is found by substituting this x back into the equation. If a > 0, the vertex is a minimum; if a < 0, it's a maximum.
Why does the quadratic formula have a ± in it?
The ± symbol accounts for both roots. Quadratic equations can have up to two solutions because a squared term (x²) can be satisfied by both a positive and negative value. The + gives the larger root and the − gives the smaller root (when the discriminant is positive).
How do I know if I should factor or use the quadratic formula?
Try factoring first if the coefficients are small integers — it's faster when it works. Use the quadratic formula when: the equation doesn't factor neatly, the coefficients are large or irrational, or you want the exact decimal values. The formula always works regardless of whether the equation is factorable.
What if a = 0 in the quadratic formula?
If a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0), solved by x = −c/b. The quadratic formula requires a ≠ 0 because the formula divides by 2a. Setting a = 0 gives division by zero, which is undefined.