The Quadratic Formula Calculator solves quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. This powerful formula can solve any quadratic equation, even when factoring is difficult or impossible, and it works for equations with real or complex solutions.

Key Features

• Automatic Solving: Simply input coefficients a, b, and c to find both solutions instantly.
• Discriminant Analysis: Calculate b² - 4ac to determine the nature of roots (two real, one real, or complex).
• Step-by-Step Solutions: See each step of the quadratic formula calculation explained clearly.
• Complex Solutions: Handle equations with complex/imaginary solutions automatically.
• Vertex Form: Convert to vertex form to identify the parabola's vertex and axis of symmetry.
• Graphical Representation: Visualize the parabola and see where it intersects the x-axis (the roots).

The Quadratic Formula

For any quadratic equation ax² + bx + c = 0 (where a ≠ 0), the solutions are given by:

The expression under the square root, b² - 4ac, is called the discriminant and determines the nature of the solutions.

Example 1: Two Real Solutions

Solve x² - 5x + 6 = 0:

Given: a = 1, b = -5, c = 6

Step 1: Calculate discriminant
Δ = b<sup>2</sup> - 4ac
Δ = (-5)<sup>2</sup> - 4(1)(6)
Δ = 25 - 24 = 1

Since Δ > 0, there are two distinct real solutions.

Step 2: Apply quadratic formula
x = (-b ± √Δ) / (2a)
x = (5 ± √1) / 2
x = (5 ± 1) / 2

Step 3: Calculate both solutions
x<sub>1</sub> = (5 + 1) / 2 = 3
x<sub>2</sub> = (5 - 1) / 2 = 2

Solutions: x = 2 or x = 3

Example 2: One Real Solution (Repeated Root)

Solve x² + 6x + 9 = 0:

Given: a = 1, b = 6, c = 9

Step 1: Calculate discriminant
Δ = 6<sup>2</sup> - 4(1)(9)
Δ = 36 - 36 = 0

Since Δ = 0, there is one repeated real solution.

Step 2: Apply quadratic formula
x = -6 / (2 × 1)
x = -6 / 2 = -3

Solution: x = -3 (with multiplicity 2)

This means the parabola touches the x-axis at exactly one point.
Factored form: (x + 3)<sup>2</sup> = 0

Example 3: Complex Solutions

Solve x² + 2x + 5 = 0:

Given: a = 1, b = 2, c = 5

Step 1: Calculate discriminant
Δ = 2<sup>2</sup> - 4(1)(5)
Δ = 4 - 20 = -16

Since Δ < 0, there are two complex conjugate solutions.

Step 2: Apply quadratic formula
x = (-2 ± √(-16)) / 2
x = (-2 ± 4i) / 2
x = -1 ± 2i

Solutions: x = -1 + 2i or x = -1 - 2i

The parabola does not intersect the x-axis.

Example 4: Coefficient a ≠ 1

Solve 2x² - 7x + 3 = 0:

Given: a = 2, b = -7, c = 3

Step 1: Calculate discriminant
Δ = (-7)<sup>2</sup> - 4(2)(3)
Δ = 49 - 24 = 25

Step 2: Apply quadratic formula
x = (7 ± √25) / (2 × 2)
x = (7 ± 5) / 4

Step 3: Calculate both solutions
x<sub>1</sub> = (7 + 5) / 4 = 12/4 = 3
x<sub>2</sub> = (7 - 5) / 4 = 2/4 = 1/2

Solutions: x = 3 or x = 1/2

Example 5: Negative Leading Coefficient

Solve -x² + 4x + 5 = 0:

Given: a = -1, b = 4, c = 5

Step 1: Calculate discriminant
Δ = 4<sup>2</sup> - 4(-1)(5)
Δ = 16 + 20 = 36

Step 2: Apply quadratic formula
x = (-4 ± √36) / (2 × -1)
x = (-4 ± 6) / -2

Step 3: Calculate both solutions
x<sub>1</sub> = (-4 + 6) / -2 = 2 / -2 = -1
x<sub>2</sub> = (-4 - 6) / -2 = -10 / -2 = 5

Solutions: x = -1 or x = 5

This parabola opens downward (a < 0).

• Standard Form: Always arrange the equation as ax² + bx + c = 0 before identifying coefficients. Move all terms to one side.
• Sign Awareness: Pay careful attention to negative signs, especially for coefficient b. If the equation is x² - 5x + 6 = 0, then b = -5 (not 5).
• Discriminant First: Calculate the discriminant (b² - 4ac) first to know what type of solutions to expect before computing them.
• Simplify Square Roots: Simplify √Δ when possible. √36 = 6, √25 = 5, √4 = 2. For negative discriminants, factor out i: √(-16) = 4i.
• Check Your Work: Substitute your solutions back into the original equation to verify they work.
• Alternative Methods: If the discriminant is a perfect square, factoring might be faster than the quadratic formula.
• Vertex Form: The x-coordinate of the vertex (axis of symmetry) is -b/(2a), which is the average of the two roots.
• Sum and Product: For roots r₁ and r₂: r₁ + r₂ = -b/a and r₁ × r₂ = c/a (Vieta's formulas).