Quadratic Formula & Factoring Calculator
Enter coefficients for ax² + bx + c = 0. This solver finds roots, shows the discriminant, and gives a factored form.
What this quadratic formula factoring calculator does
This tool combines three common algebra tasks in one place:
- Solves a quadratic equation using the quadratic formula
- Calculates the discriminant to classify the roots
- Writes the equation in factored form (real or complex form when needed)
If you are learning algebra, precalculus, or reviewing for tests like SAT/ACT, this calculator helps you check your work quickly and understand each result.
Quadratic formula refresher
A quadratic equation is usually written as:
The roots are found by:
The expression inside the square root, b² - 4ac, is called the discriminant.
How to read the discriminant
- D > 0: two different real roots
- D = 0: one repeated real root
- D < 0: two complex conjugate roots
How factoring connects to the roots
Once roots are known, factoring becomes straightforward. If roots are r1 and r2, then:
That means the quadratic formula is not just for solving equations—it also gives you the exact values needed for factoring.
How to use this calculator
- Type values for a, b, and c.
- Click Calculate.
- Read the equation form, discriminant, roots, and factorization.
- Click Clear to start over.
Tip: a cannot be 0, or the equation is not quadratic.
Example problems
Example 1: Easy factoring case
For x² - 5x + 6 = 0, the roots are 2 and 3. Factored form: (x - 2)(x - 3).
Example 2: Repeated root
For x² - 4x + 4 = 0, the discriminant is 0, so the root is repeated at 2. Factored form: (x - 2)².
Example 3: Complex roots
For x² + 2x + 5 = 0, the discriminant is negative. Roots are complex, so factoring over the real numbers is not possible.
Common mistakes to avoid
- Forgetting the parentheses in -b and 2a
- Sign errors when b is negative
- Treating a negative discriminant as an error (it just means complex roots)
- Dropping the leading coefficient a in the final factored form
Final thoughts
A strong quadratic equation solver should do more than output two numbers. It should explain whether roots are real or complex, show the discriminant, and connect results to factoring. Use this page whenever you need a fast, clear quadratic formula calculator and factoring helper in one step.