Quadratic Formula Calculator – Solve ax² + bx + c = 0
Quadratic Formula Solver
Equation: ax² + bx + c = 0
The Quadratic Formula Calculator solves any quadratic equation of the form ax² + bx + c = 0. It computes the discriminant, finds real or complex roots, and provides a complete step‑by‑step explanation. The quadratic formula is one of the most important tools in algebra, used in physics, engineering, finance, and many other fields. Whether you need to find the x‑intercepts of a parabola, solve projectile motion problems, or optimise profit functions, this calculator gives instant, accurate results.
Quadratic Formula
x = [-b ± √(b² - 4ac)] / (2a)
The Discriminant
Δ = b² – 4ac
• If Δ greater than 0 → two distinct real roots (parabola crosses x-axis twice)
• If Δ = 0 → one real double root (vertex touches x-axis)
• If Δ less than 0 → two complex conjugate roots (no real x-intercepts)
Real‑World Applications
Physics: Projectile motion (height vs. time), free‑fall trajectory, rocket propulsion.
Geometry: Finding dimensions of shapes given area or perimeter constraints.
Derivation of the Quadratic Formula
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. Divide by a, move the constant term, add (b/2a)² to both sides, factor the left, then take the square root. The result is the familiar formula. This derivation works for all a, b, c (a ≠ 0).
Discriminant Summary Table (Δ = b² - 4ac)
Discriminant Value
Nature of Roots
Parabola X-Intercepts
Example Equation
Δ > 0 (Positive)
2 Distinct Real Roots
Crosses x-axis twice
x² - 5x + 6 = 0 (x = 2, 3)
Δ = 0 (Zero)
1 Real Double Root
Vertex touches x-axis
x² + 4x + 4 = 0 (x = -2)
Δ < 0 (Negative)
2 Complex Conjugate Roots
Does not intersect x-axis
x² + x + 1 = 0 (x = -0.5 ± 0.866i)
Comparing Methods for Solving Quadratic Equations
Solving Method
Best Applicable When
Key Advantage
Limitations
Quadratic Formula
Any quadratic equation ax² + bx + c = 0
100% universal; solves all real and complex roots.
Involves longer multi-step arithmetic.
Factoring (FOIL)
Simple integer coefficients with rational roots
Fastest method when factors are obvious.
Fails for non-factorable equations or imaginary roots.
Completing the Square
When a = 1 and coefficient b is an even integer
Converts standard form directly into vertex form.
Creates complex fractions when a ≠ 1 or b is odd.
Square Root Property
Equations missing linear term b = 0 (ax² + c = 0)
Directly isolates x² and takes square root.
Only applies when b = 0.
Common Mistakes When Using the Quadratic Formula
Forgetting the ± sign: Both addition and subtraction roots must be computed.
Mis‑identifying signs of b and c: Always rewrite your equation into standard form ax² + bx + c = 0 before extracting coefficients.
Incorrect calculation of discriminant: Compute b² – 4ac, taking care that (-b)² is always positive.
Dividing by a only after subtracting c: The entire numerator [-b ± √(b² - 4ac)] must be divided by 2a at the very end.
Setting a = 0: If a = 0, the equation reduces to a linear equation bx + c = 0, so the quadratic formula cannot be applied.
How to Derive the Quadratic Formula (Step‑by‑Step)
Start with ax² + bx + c = 0 (a ≠ 0).
Divide both sides by a: x² + (b/a)x + c/a = 0.
Move constant term: x² + (b/a)x = -c/a.
Complete the square: add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)².
Left side is (x + b/2a)² = (b² - 4ac)/4a².
Take square root: x + b/2a = ±√(b² - 4ac) / (2a).
Solve for x: x = [-b ± √(b² - 4ac)] / (2a).
Use this quadratic formula calculator to check your homework, verify your work, or explore how changing coefficients affects the roots. The step‑by‑step output makes it an excellent learning tool for students.
Step‑by‑Step Manual Examples
Example 1: Two Real Distinct Roots (x² – 5x + 6 = 0)
Example 3: Complex Conjugate Roots (x² + x + 1 = 0)
Identify a = 1, b = 1, c = 1
Discriminant Δ = 1² – 4×1×1 = 1 – 4 = -3 (Δ < 0)
Real part = -1 / (2×1) = -0.5
Imaginary part = √3 / 2 ≈ 0.866025
Roots: x = -0.5 ± 0.866025i
Frequently Asked Questions about Quadratic Equations
What is the quadratic formula?
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). It provides the exact solutions (roots or x-intercepts) for any second-degree polynomial equation in standard form ax² + bx + c = 0.
What is the discriminant in the quadratic formula?
The discriminant is Δ = b² - 4ac (the expression underneath the square root). It determines the nature and number of roots: Δ > 0 gives 2 distinct real roots, Δ = 0 gives 1 real double root, and Δ < 0 gives 2 complex conjugate roots.
How do you solve quadratic equations with complex imaginary roots?
When the discriminant Δ < 0, taking the square root yields an imaginary number i = √(-1). The formula evaluates to x = -b/(2a) ± [√(-Δ)/(2a)]i, resulting in two complex conjugate numbers.
Can coefficient 'a' be zero in a quadratic equation?
No. If a = 0, the x² term vanishes, transforming the equation into a linear equation (bx + c = 0) rather than a quadratic one. Therefore, a quadratic equation requires a ≠ 0.
How does the quadratic formula relate to a parabola's graph?
Graphically, the quadratic function y = ax² + bx + c forms a U-shaped parabola. The roots solved by the formula correspond to the x-intercepts where y = 0. The formula's first part (-b / 2a) gives the x-coordinate of the parabola's vertex.
How is the quadratic formula derived?
The formula is algebraically derived by taking the standard equation ax² + bx + c = 0 and completing the square. After dividing by 'a', adding (b/2a)² to both sides, and taking the square root, you get x = [-b ± √(b² - 4ac)] / (2a).
When should I use the quadratic formula versus factoring?
Factoring is faster when coefficients are simple small integers with obvious factors. However, the quadratic formula works universally for every quadratic equation, including non-factorable equations, decimals, and equations with complex roots.
Can I input decimal or negative coefficients into this calculator?
Yes! Our calculator accepts any positive or negative real numbers, including decimal coefficients like a = 1.5, b = -3.2, and c = 0.75.