How to Solve a Quadratic Equation Step by Step
A quadratic equation is any equation that can be written in the standard form ax2 + bx + c = 0, where a, b, and c are numbers (constants) and a is not zero. Solving it means finding the value or values of x that make the equation true. Here is exactly how to do that using the quadratic formula.
The Quadratic Formula
For any equation in the form ax2 + bx + c = 0, the solutions for x are given by:
x = (-b +/- sqrt(b2 - 4ac)) / 2a
The +/- symbol means there are generally two solutions - one using addition, one using subtraction - since a quadratic equation can have up to two values of x that satisfy it.
Step 1: Identify a, b, and c
Before using the formula, write your equation in standard form and identify each coefficient. For example, in the equation:
2x2 - 4x - 6 = 0
a = 2, b = -4, c = -6
Step 2: Calculate the Discriminant
The discriminant is the expression inside the square root: b2 - 4ac. Its value tells you what type of solutions to expect, before you even finish solving:
- Positive discriminant: Two distinct real solutions
- Zero discriminant: Exactly one real solution (a repeated root)
- Negative discriminant: No real solutions - the two solutions are complex numbers
Using our example: Discriminant = (-4)2 - 4(2)(-6) = 16 + 48 = 64
Since 64 is positive, we know there will be two distinct real solutions.
Step 3: Apply the Quadratic Formula
x = (-(-4) +/- sqrt(64)) / (2 x 2)
x = (4 +/- 8) / 4
This gives two solutions:
x1 = (4 + 8) / 4 = 12 / 4 = 3
x2 = (4 - 8) / 4 = -4 / 4 = -1
Step 4: Check Your Answer (Optional but Recommended)
You can verify a solution by substituting it back into the original equation. Checking x = 3:
2(3)2 - 4(3) - 6 = 2(9) - 12 - 6 = 18 - 12 - 6 = 0 (correct)
Since the equation equals zero, x = 3 is confirmed correct.
What If the Discriminant Is Negative?
When b2 - 4ac is negative, there is no real number whose square gives that negative value, so the solutions involve the imaginary unit i (where i2 = -1). These are called complex roots, and they always come in a conjugate pair (one with +i, one with -i) for real-coefficient quadratic equations.
A Note on the Vertex
Every quadratic equation, when graphed, forms a parabola. The vertex (the highest or lowest point of that parabola) can be found using:
x-coordinate of vertex = -b / 2a
This is useful for understanding the shape of the equation beyond just its roots, and it is actually part of the derivation of the quadratic formula itself.
Common Mistakes to Avoid
- Forgetting the +/- sign - most quadratics have two solutions, not one.
- Sign errors with negative coefficients - be careful when b or c is negative, since -b and -(-b) are easy to mix up.
- Not simplifying the square root fully when the discriminant is not a perfect square.
Skip the Manual Math - Use Our Free Quadratic Equation Solver
Our free Quadratic Equation Solver instantly calculates the discriminant, both roots (real or complex), and the vertex for any equation in the form ax2 + bx + c = 0 - just enter your coefficients.
Final Thoughts
Solving a quadratic equation always follows the same reliable process: identify a, b, and c, calculate the discriminant to know what kind of solutions to expect, then apply the quadratic formula. Once you have worked through a few examples by hand, the pattern becomes straightforward - and a calculator is there for when you just need the answer quickly.