Quadratic Formula Calculator
Solve any quadratic equation with every step shown. Enter a, b and c (fractions are fine) or type the equation, and get exact and decimal roots, the discriminant, factored form, vertex and a graph.
Whole numbers, decimals or fractions: 3, −2.5, 3/4 or −1 1/2. Use 0 for a missing term.
Type x^2 or x² for x squared. Brackets, fractions and terms on both sides are fine: everything is moved to one side for you.
Pick >, ≥, < or ≤ to solve an inequality.
How your roots are worked out
Your numbers, step by step: read off a, b and c, find the discriminant, then use the quadratic formula.
- 1
Find a, b and c
a = 2, b = −4c = −62x² − 4x − 6 = 0
- 2
Work out b² − 4ac
(−4)² − 4·2·(−6)= 16 + 48= 64
- 3
Use the formula
x = (−b ± √D) / 2a= (4 ± √64) / 4√64 = 8
- 4
Your two roots
x = 3x = −1two rational roots
Read more: the formula, a worked example, completing the square and factoring
A quadratic equation has an x² term and no higher power: ax² + bx + c = 0, where a is not 0. Its graph is a U-shaped curve called a parabola. The roots (also called solutions or zeros) are the x values where the curve crosses the x-axis. The quadratic formula finds them for any a, b and c. The discriminant, b² − 4ac, tells you before you finish how many real roots there are.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a
discriminant D = b² − 4ac (D > 0: two real roots, D = 0: one repeated root, D < 0: two complex roots)
vertex: x = −b ÷ 2a, y = c − b² ÷ 4a
vertex form: y = a(x − h)² + k, where (h, k) is the vertex
sum of roots = −b ÷ a, product of roots = c ÷ a
Worked example
Solve 2x² − 4x − 6 = 0. Here a = 2, b = −4 and c = −6. The discriminant is (−4)² − 4 × 2 × (−6) = 16 + 48 = 64. That is positive and a perfect square (8 × 8), so there are two rational roots. The formula gives x = (4 ± √64) ÷ (2 × 2) = (4 ± 8) ÷ 4. With plus: 12 ÷ 4 = 3. With minus: −4 ÷ 4 = −1. Check: 2 × 3² − 4 × 3 − 6 = 18 − 12 − 6 = 0. So the factored form is 2(x − 3)(x + 1). The vertex is at x = 4 ÷ 4 = 1, where y = 2 − 4 − 6 = −8, so the vertex is (1, −8) and the vertex form is 2(x − 1)² − 8. The graph crosses the y-axis at (0, −6).
Completing the square, step by step
Divide 2x² − 4x − 6 = 0 by 2: x² − 2x − 3 = 0. Move the 3 across: x² − 2x = 3. Half of −2 is −1, and (−1)² = 1, so add 1 to both sides: x² − 2x + 1 = 4. The left side is (x − 1)², so (x − 1)² = 4. Take the square root of both sides, remembering both signs: x − 1 = ±2, so x = 3 or x = −1. This is exactly how the quadratic formula is proved: complete the square on ax² + bx + c = 0 and you get x = (−b ± √(b² − 4ac)) ÷ 2a.
Factoring with the ac method
Multiply a × c (2 × −6 = −12) and look for two numbers that multiply to it and add to b (−4): 2 and −6. Split the middle term: 2x² + 2x − 6x − 6. Group in pairs: 2x(x + 1) − 6(x + 1) = (2x − 6)(x + 1). Each bracket can be 0, so x = 3 or x = −1. This only works neatly when the discriminant is a perfect square. Otherwise use the formula.
What the discriminant tells you
| b² − 4ac | Roots | Graph | Example |
|---|---|---|---|
| Positive, perfect square | Two rational (factors neatly) | Crosses the x-axis twice | x² + x − 6: D = 25, x = 2, −3 |
| Positive, not a square | Two irrational (surds) | Crosses the x-axis twice | x² − 12: D = 48, x = ±2√3 |
| Zero | One repeated root | Touches the x-axis at the vertex | x² − 4x + 4: D = 0, x = 2 |
| Negative | Two complex (a ± bi) | Never touches the x-axis | x² + 2x + 5: D = −16, x = −1 ± 2i |
Simplifying the square root (surds)
Take out the largest square factor: √48 = √(16 × 3) = 4√3, and √72 = √(36 × 2) = 6√2. Then cancel any number that divides the top and bottom: (−6 ± 2√5) ÷ 2 = −3 ± √5. For a negative discriminant, √−16 = √16 × √−1 = 4i.
Inequalities
To solve 2x² − 4x − 6 > 0, find the roots (−1 and 3). The parabola opens up, so it is above the x-axis outside the roots: x < −1 or x > 3, which is (−∞, −1) ∪ (3, ∞) in interval notation. For < 0 it is between them: −1 < x < 3. Use ≥ or ≤ to include the roots themselves (square brackets).
Typing an equation
- x squared can be x^2, x² or x*x. Leave out the ×: 3x(x − 2) works.
- Terms can be on both sides: 2x² − 4x = 6 becomes 2x² − 4x − 6 = 0.
- Brackets are multiplied out: (x − 2)(x + 3) = 0 becomes x² + x − 6 = 0.
- 1/2x² means one half times x². y = … and f(x) = … are solved where y = 0.
- Any one letter works as the unknown, like t² + 2t = 8.
All answers are worked out exactly with whole numbers and fractions, so the surds and fractions shown are exact. Decimals are rounded to the places you choose.
Common questions
What is the quadratic formula?
For any equation ax² + bx + c = 0 with a not 0, x = (−b ± √(b² − 4ac)) / 2a. The ± gives two answers: one with plus and one with minus. For 2x² − 4x − 6 = 0, a = 2, b = −4 and c = −6, so x = (4 ± √64) / 4 = (4 ± 8) / 4, which gives x = 3 or x = −1.
What does the discriminant tell you?
The discriminant is b² − 4ac, the part under the square root. If it is positive there are two different real roots, and if it is also a perfect square (like 64 = 8²) they are whole numbers or fractions and the quadratic factors neatly. If it is 0 there is one repeated root. If it is negative there are no real roots, only two complex ones.
How do you find the vertex of a parabola?
The vertex is on the axis of symmetry, x = −b / 2a. Put that x back into the equation to get its height. For y = 2x² − 4x − 6, x = 4 / 4 = 1 and y = 2 − 4 − 6 = −8, so the vertex is (1, −8). Because a is positive the parabola opens up, so this is its lowest point.
How do you solve a quadratic by completing the square?
Divide by a, move the number to the right, then add (half the x coefficient)² to both sides so the left side becomes a perfect square. For 2x² − 4x − 6 = 0: x² − 2x = 3, add 1 to get (x − 1)² = 4, so x − 1 = ±2 and x = 3 or x = −1.
What are complex roots?
When the discriminant is negative, its square root is an imaginary number, written with i = √−1. For x² + 2x + 5 = 0 the discriminant is 4 − 20 = −16 and √−16 = 4i, so x = (−2 ± 4i) / 2 = −1 ± 2i. The graph never touches the x-axis.
Can I use fractions or decimals for a, b and c?
Yes. Type 3/4, −1 1/2 or 0.25 in any box. The calculator multiplies every term by the lowest common denominator first, which keeps the same roots. For 1/2x² + 1/3x − 1 = 0 it multiplies by 6 to get 3x² + 2x − 6 = 0, so x = (−1 ± √19) / 3.
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