Quadratic Formula Calculator
Solve any quadratic equation ax²+bx+c=0 instantly. Enter the coefficients a, b, and c to find real or complex roots, the discriminant, vertex coordinates, and full step-by-step working. Covers all cases: two roots, double root, and complex roots.
Enter the coefficients for ax² + bx + c = 0
Use negative values for negative coefficients (e.g. b = −5). The coefficient a must not be zero.
The Quadratic Formula
The quadratic formula is one of the most important tools in algebra. It solves any quadratic equation of the form ax² + bx + c = 0, where a ≠ 0. The formula is:
Where Δ = b² − 4ac is the discriminant
The ± symbol means there are generally two solutions: one with the + sign (x₁) and one with the − sign (x₂). Whether these are real or complex depends on the discriminant.
Deriving the Formula by Completing the Square
The quadratic formula is not arbitrary — it is derived from the technique of completing the square. Here is the derivation from ax² + bx + c = 0:
Derivation of the Quadratic Formula
Step 1: Divide all terms by a:
Step 2: Move the constant to the right:
Step 3: Complete the square — add (b/2a)² to both sides:
Step 4: Left side is a perfect square trinomial:
Step 5: Take the square root:
Step 6: Solve for x:
The Discriminant: Δ = b² − 4ac
Before solving, calculate the discriminant Δ = b² − 4ac. Its value tells you the nature of the roots:
| Discriminant Δ | Number of Real Roots | Graphical Meaning | Root Type |
|---|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis at 2 points | Real, different |
| Δ = 0 | One repeated root (double root) | Parabola touches x-axis at 1 point | Real, equal |
| Δ < 0 | No real roots | Parabola does not cross x-axis | Complex conjugates |
Three Methods for Solving Quadratics
There are three main methods. Knowing when to use each saves time in exams:
| Method | Best Used When | Always Works? |
|---|---|---|
| Factoring | Small integer roots, equation factors neatly | No (only when factorable over integers) |
| Completing the Square | Finding vertex form, deriving the formula | Yes (but algebraically intensive) |
| Quadratic Formula | Any quadratic, especially non-integer roots | Yes — always works for any quadratic |
Worked Examples: All Three Cases
Example 1: Two Distinct Real Roots — x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Step 1 — Discriminant:
Step 2 — Apply formula:
Step 3 — Two solutions:
Factored form: (x − 3)(x − 2) = 0. Check: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓
Example 2: Double Root — x² − 6x + 9 = 0
a = 1, b = −6, c = 9
Factored form: (x − 3)² = 0. The parabola is tangent to the x-axis at x = 3.
Example 3: Complex Roots — x² + x + 1 = 0
a = 1, b = 1, c = 1
Roots: x = −0.5 + 0.866i and x = −0.5 − 0.866i. These are complex conjugates. The parabola does not cross the x-axis.
Vertex Form and the Parabola
Every quadratic equation y = ax² + bx + c describes a parabola. The vertex (turning point) is the minimum point when a > 0 (opens upward) or maximum when a < 0 (opens downward).
The x-coordinate of the vertex is the axis of symmetry; the y-coordinate is the minimum or maximum value
The vertex form of a quadratic is y = a(x − h)² + k, where (h, k) is the vertex. Converting to vertex form is equivalent to completing the square.
Vieta’s Formulas: Sum and Product of Roots
For the quadratic ax² + bx + c = 0 with roots x₁ and x₂, Vieta’s formulas give:
- Sum of roots: x₁ + x₂ = −b/a
- Product of roots: x₁ × x₂ = c/a
These are useful for checking answers and for quickly finding quadratics with given roots. For example, to find the quadratic with roots 3 and 2: sum = 5 so b/a = −5; product = 6 so c/a = 6 → x² − 5x + 6 = 0.
Applications of Quadratics
Example 4: Projectile Motion
A ball is thrown upward with initial velocity 20 m/s from a height of 2 m. When does it hit the ground?
Using h(t) = −5t² + 20t + 2 = 0 (using g = 10 m/s²):
a = −5, b = 20, c = 2. Δ = 400 + 40 = 440
Taking the positive root: t = (−20 − 20.976) / (−10) ≈ 4.10 seconds
Example 5: Area Problem
A rectangular garden has a length 3 metres more than its width. The area is 40 m². Find the dimensions.
Let width = x. Then length = x + 3 and area = x(x + 3) = 40 → x² + 3x − 40 = 0
Δ = 9 + 160 = 169. x = (−3 ± 13) / 2
x = 5 (taking positive root). Width = 5m, length = 8m. Check: 5 × 8 = 40 ✓
Completing the Square: Step-by-Step
Completing the square is used to solve quadratics, convert to vertex form, and understand graphical properties:
Example 6: Complete the square for x² + 6x + 5 = 0
Step 1: Move constant: x² + 6x = −5
Step 2: Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4
Step 3: Factor left: (x + 3)² = 4
Step 4: Take square root: x + 3 = ±2
Solutions: x = −3 + 2 = −1 or x = −3 − 2 = −5. Vertex is at (−3, −4).
Quadratic Equations in GCSE and A-Level Maths
At GCSE (Higher tier), quadratic equation questions require:
- Solving by factoring (for factorable quadratics)
- Applying the quadratic formula to non-integer solutions (usually to 2 or 3 s.f.)
- Completing the square to find vertex or solve
At A-Level, quadratics appear in more advanced contexts:
- Quadratics in disguise: e.g. substituting u = sin x to get 2u² − 3u + 1 = 0
- Quadratic inequalities: solving ax² + bx + c > 0 by finding roots and using sign diagrams
- Discriminant conditions: for what range of k does kx² − 3x + 2 = 0 have real roots?
Example 7 (A-Level): Find the range of k for which x² + kx + 4 = 0 has real roots
For real roots, Δ ≥ 0:
Example 8: Quadratic in disguise — Solve 2sin²x − sin x − 1 = 0, 0 ≤ x ≤ 360°
Let u = sin x: 2u² − u − 1 = 0 → (2u + 1)(u − 1) = 0
u = −1/2 or u = 1 → sin x = −0.5 or sin x = 1
Solutions: x = 210°, 330°, 90°
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic equation ax² + bx + c = 0, where a ≠ 0. The formula gives two solutions because of the ± sign. These can be two real numbers (when the discriminant Δ = b² − 4ac > 0), one repeated number (when Δ = 0), or two complex numbers (when Δ < 0). The formula was known to ancient Babylonian and Greek mathematicians, though the modern algebraic form was developed by Islamic mathematician Al-Khwarizmi around 820 AD.
What is the discriminant and what does it tell you?
The discriminant Δ = b² − 4ac is the expression under the square root in the quadratic formula. It tells you the nature of the roots before you solve: if Δ > 0 there are two distinct real roots; if Δ = 0 there is exactly one repeated root (the parabola touches the x-axis); if Δ < 0 there are no real roots (only complex roots). At A-Level, you often need to set conditions on the discriminant: for example, "for what values of k does the equation have no real solutions?" requires Δ < 0.
When should you use the quadratic formula vs factoring?
Use factoring when the equation factors neatly with small integers (e.g. x² + 5x + 6 = (x+2)(x+3)). It is faster when it works. Use the quadratic formula when: the discriminant is not a perfect square, the roots are decimals or fractions, or the equation resists obvious factoring. Use completing the square when you need the vertex form y = a(x−h)² + k. The quadratic formula always works and is the safest method under time pressure.
How do you complete the square?
For ax² + bx + c = 0: (1) Divide all terms by a. (2) Move the constant to the right. (3) Add (b/2a)² to both sides to create a perfect square on the left. (4) The left side is (x + b/2a)². (5) Take the square root and solve for x. For example, x² + 4x + 1 = 0 → x² + 4x = −1 → (x+2)² = 3 → x = −2 ± √3 ≈ 0.732 or −3.732.
What is a double root?
A double root (or repeated root) occurs when the discriminant Δ = 0. Both solutions of the quadratic formula give the same value: x = −b/(2a). This means the quadratic can be written as a(x − r)² = 0, where r is the repeated root. Graphically, the parabola is tangent to the x-axis: it just touches the x-axis at exactly one point without crossing it. Example: x² − 4x + 4 = (x−2)² = 0 has a double root at x = 2.
Can a quadratic have no real solutions?
Yes — when Δ = b² − 4ac < 0. The square root of a negative number is imaginary, so the roots are complex: x = −b/(2a) ± i√|Δ|/(2a), where i = √(−1). For example, x² + 2x + 5 = 0 has Δ = 4−20 = −16, giving roots x = −1 ± 2i. On a graph, the parabola lies entirely above the x-axis (if a > 0) and never crosses it. At A-Level, complex roots always come in conjugate pairs (a+bi and a−bi).