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Quadratic Formula Calculator

Enter the coefficients a, b, and c for the equation ax² + bx + c = 0. The calculator finds all roots (real or complex), the discriminant, and the vertex, with full step-by-step working.

Equation: ax² + bx + c = 0

Results

x₁:3
x₂:2
Discriminant (D):1
Axis of symmetry:x = 2.5
Vertex:(2.5, -0.25)
Step-by-step work:
Equation: 1x² -5x +6 = 0

Step 1 — Identify coefficients:
  a = 1,  b = -5,  c = 6

Step 2 — Quadratic formula:
  x = (-b ± √(b² - 4ac)) / 2a

Step 3 — Calculate the discriminant:
  D = b² - 4ac
  D = (-5)² - 4 × 1 × 6
  D = 25 - 24
  D = 1

Step 4 — Evaluate √D:
  √|D| = √1 = 1

Step 5 — Substitute into formula:
  x = (-(-5) ± 1) / (2 × 1)
  x = (5 ± 1) / 2

  D > 0 → Two distinct real roots:
  x₁ = (5 + 1) / 2 = 3
  x₂ = (5 - 1) / 2 = 2

Step 6 — Vertex and axis of symmetry:
  Axis of symmetry: x = -b / 2a = 5 / 2 = 2.5
  Vertex y = 1(2.5)² + -5(2.5) + 6 = -0.25
  Vertex: (2.5, -0.25)

The Quadratic Formula

A quadratic equation is any equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The quadratic formula gives the solution directly:

x = (−b ± √(b² − 4ac)) / (2a)

The ± symbol produces two solutions: one with a plus sign and one with a minus sign. These are called the two roots of the equation. They represent the x-values where the parabola y = ax² + bx + c crosses the x-axis.

Deriving the Formula: Completing the Square

The quadratic formula is not a magic rule — it comes directly from a technique called completing the square. Starting from the general form:

Step 1: ax² + bx + c = 0
Step 2: Divide by a: x² + (b/a)x = −c/a
Step 3: Add (b/2a)² to both sides:
  x² + (b/a)x + (b/2a)² = (b/2a)² − c/a
Step 4: Factor left side: (x + b/2a)² = (b² − 4ac) / 4a²
Step 5: Take square root: x + b/2a = ±√(b² − 4ac) / 2a
Step 6: Solve for x: x = (−b ± √(b² − 4ac)) / 2a

Completing the square is also how you convert between standard form (ax² + bx + c) and vertex form (a(x − h)² + k).

The Discriminant: Predicting the Roots

The expression under the square root sign, D = b² − 4ac, is called the discriminant. It tells you the nature of the roots before you finish the calculation:

Discriminant value Number of roots Type of roots
D > 0 Two Two distinct real roots
D = 0 One One repeated real root (tangent to x-axis)
D < 0 Two Two complex conjugate roots (no real solutions)

When D < 0, the roots are complex numbers involving the imaginary unit i = √(−1). The two roots take the form p + qi and p − qi, where p = −b/(2a) and q = √(−D)/(2a).

Step-by-Step Example

Solve: 2x² − 7x + 3 = 0

a = 2, b = −7, c = 3.
Step 1 — Discriminant: D = (−7)² − 4(2)(3) = 49 − 24 = 25. D > 0, two real roots.
Step 2 — Apply formula: x = (7 ± √25) / (2 × 2) = (7 ± 5) / 4
Step 3 — Two roots:
  x&sub1; = (7 + 5) / 4 = 12 / 4 = 3
  x&sub2; = (7 − 5) / 4 = 2 / 4 = 0.5
Check: 2(3)² − 7(3) + 3 = 18 − 21 + 3 = 0. Correct.

The Vertex of the Parabola

The graph of y = ax² + bx + c is a parabola. Its vertex (the turning point) has an x-coordinate exactly halfway between the two roots:

xvertex = −b / (2a)

Substitute this back into the equation to find the y-coordinate. The vertex form of the parabola is y = a(x − h)² + k, where (h, k) is the vertex. If a > 0, the parabola opens upward and the vertex is the minimum. If a < 0, it opens downward and the vertex is the maximum.

Real-World Applications

Quadratic equations model a wide range of real phenomena:

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0 (where a ≠ 0). The formula is: x = (−b ± √(b² − 4ac)) / (2a). The ± symbol means there are potentially two solutions. The expression under the square root, b² − 4ac, is called the discriminant. The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0.

What does the discriminant tell you?

The discriminant D = b² − 4ac determines the nature and number of roots: If D > 0, the equation has two distinct real roots. If D = 0, the equation has exactly one real root (a repeated root). If D < 0, the equation has two complex (imaginary) roots — the square root of a negative number introduces the imaginary unit i. The discriminant lets you predict the type of solution before calculating it.

How do you complete the square to derive the quadratic formula?

Start with ax² + bx + c = 0. Divide through by a: x² + (b/a)x + c/a = 0. Move the constant: x² + (b/a)x = −c/a. Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = (b/2a)² − c/a. Factor left side: (x + b/2a)² = (b² − 4ac)/4a². Take square root: x + b/2a = ±√(b² − 4ac)/(2a). Solve for x: x = (−b ± √(b² − 4ac))/(2a).

What is the vertex of a parabola and how do you find it?

The vertex of the parabola y = ax² + bx + c is the point where the curve reaches its minimum (if a > 0) or maximum (if a < 0). The x-coordinate of the vertex is x = −b/(2a). This is exactly the midpoint between the two roots when they exist — and the formula comes directly from the quadratic formula by setting the ± part to zero. The y-coordinate is found by substituting x = −b/(2a) back into the original equation.

When should you use factoring instead of the quadratic formula?

Factoring is faster when the coefficients are small integers and the equation factors neatly. For example, x² − 5x + 6 = 0 factors to (x − 2)(x − 3) = 0 immediately. But factoring is not always possible or obvious. The quadratic formula always works for any quadratic equation, regardless of whether it factors. If the discriminant is not a perfect square, factoring over integers is impossible and the formula is the right tool.

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