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Quadratic Formula Calculator
Solve any quadratic equation of the form ax² + bx + c = 0 and see whether it has two real roots, one repeated root or complex roots.
About the Quadratic Formula Calculator
Solve any quadratic equation of the form ax² + bx + c = 0 and see whether it has two real roots, one repeated root or complex roots.
How to use the Quadratic Formula Calculator
- Enter the coefficient a of x², the coefficient b of x and the constant c.
- Press Solve Equation to see the roots and the discriminant.
Formula and method
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant D = b² − 4ac tells you the type of roots: D > 0 gives two real roots, D = 0 gives one repeated root and D < 0 gives two complex roots.
Worked example
For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6. D = 25 − 24 = 1, so x = (5 ± 1) ÷ 2, which gives x = 3 and x = 2.
Solving x² − 5x + 6 = 0. The discriminant is (−5)² − 4·1·6 = 1, so x = (5 ± 1) ÷ 2, giving x = 3 and x = 2.
When to use the Quadratic Formula Calculator
- Solving quadratic equations in algebra homework.
- Finding when a thrown object hits the ground or reaches a height.
- Checking whether an equation has real solutions.
Common mistakes to avoid
- Forgetting to move everything to one side so the equation equals zero.
- Getting a sign wrong when b is negative.
- Dividing only part of the numerator by 2a.
Tips
- The value of a cannot be zero; that would make the equation linear.
- Check answers by substituting them back into the original equation.
Frequently asked questions
What are complex roots?
When the discriminant is negative there are no real solutions. The roots include the imaginary unit i, where i² = −1.
What does the discriminant tell me?
If b² − 4ac is positive there are two real solutions, if it is zero there is one, and if it is negative there are no real solutions.
Can I solve without the formula?
Sometimes by factorising or completing the square, but the formula always works.