Factorable quadratic
Solve x^2-5x+6=0
Method: Factor using product and sum.
- 1Find -2 and -3 because their product is 6 and sum is -5.
- 2Write (x-2)(x-3)=0.
- 3Set each factor equal to zero.
Final answer
Free online calculator
Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using factoring, completing the square, or the quadratic formula.
Step-by-step solutions
Clear, detailed explanations
Instant results
Accurate answers in seconds
Free to use
No sign-up required
How to Use the Quadratic Formula Calculator
1
Enter the quadratic
Type ax² + bx + c = 0 in any form.
2
Choose a method
We factor when possible; otherwise we use the quadratic formula.
3
See the roots
Get real or complex roots with verification.
Example
How the Quadratic Formula Calculator Chooses a Method
Find all roots of a quadratic and understand whether factoring, completing the square, or the quadratic formula is appropriate.
The solver converts the equation to ax^2+bx+c=0, checks whether it factors cleanly, and otherwise applies the quadratic formula. The discriminant determines how many real roots exist.
Supported problem types
- Quadratics with real or complex roots
- Factoring and quadratic-formula solutions
- Exact radical and decimal root forms
Limits and assumptions
- The leading coefficient a must be nonzero for the equation to be quadratic.
- Decimal coefficients can make an exact factored form less convenient.
Worked Quadratic Formula Calculator Examples
Each example names the method, shows the ordered reasoning, and keeps the final answer separate so you can check your own work.
Repeated root
Solve x^2-6x+9=0
Method: Recognize a perfect-square trinomial.
- 1The expression factors as (x-3)^2.
- 2Set x-3 equal to zero.
- 3The discriminant is zero, confirming one repeated root.
Final answer
Complex roots
Solve x^2+4=0
Method: Isolate x^2 and use i^2=-1.
- 1Move 4 to obtain x^2=-4.
- 2Take both square roots.
- 3Write sqrt(-4) as 2i.
Final answer
Learn This Topic
What the Quadratic Formula Calculator Helps You Understand
Solve any quadratic equation ax² + bx + c = 0 using factoring, completing the square, or the quadratic formula.
Learning focus 1
Type ax² + bx + c = 0 in any form.
Learning focus 2
We factor when possible; otherwise we use the quadratic formula.
Learning focus 3
Get real or complex roots with verification.
Input tips
- • Type ax² + bx + c = 0 in any form.
- • Yes. If the discriminant is negative, the solver returns complex roots in a + bi form.
Common mistakes to avoid
- • Dividing by an expression that may be zero.
- • Keeping extraneous roots introduced by squaring or clearing denominators.
Frequently Asked Questions
Can it handle complex roots?
Yes. If the discriminant is negative, the solver returns complex roots in a + bi form.
Does it show the discriminant?
Yes. The solver computes and explains the discriminant and what it means for the number of roots.
What does the discriminant tell me?
A positive discriminant gives two real roots, zero gives one repeated real root, and a negative value gives two complex roots.
Why does the quadratic formula always work?
It comes from completing the square on the general equation ax^2+bx+c=0, so it applies to every quadratic with a nonzero a.