Quadratic Calculator

Solve quadratic equations, calculate discriminant, find vertex, and convert to factored form with our free quadratic calculator

Available Calculators

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, typically written as ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. Quadratic equations are fundamental in algebra and appear in many real-world applications, from physics to economics.

Quadratic Formula

The quadratic formula is used to solve quadratic equations:

x = ((-b ± √(b^2 - 4ac)) ÷ 2a)

Discriminant

The discriminant (Δ = b² - 4ac) determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; if Δ < 0, there are two complex roots.

Examples of Quadratic Calculations

Here are several everyday situations that show how easy it is to use our quadratic calculator.

Example 1: Solve x² - 5x + 6 = 0

Solve the quadratic equation x² - 5x + 6 = 0.

You can use our calculator by entering:

What are the roots of x² - 5x + 6 = 0?

The calculator returns: x = 2, x = 3

a = 1, b = -5, c = 6
Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
x = (5 ± 1) / 2
x₁ = 3, x₂ = 2

The formula used is:

x = ((5 ± √(25 - 24)) ÷ 2) = ((5 ± 1) ÷ 2)

Example 2: Discriminant Analysis

Calculate the discriminant of x² + 4x + 4 = 0.

You can use our calculator by entering:

What is the discriminant of x² + 4x + 4 = 0?

The calculator returns: Δ = 0 (one repeated root)

Δ = b² - 4ac
Δ = 4² - 4(1)(4)
Δ = 16 - 16 = 0
Since Δ = 0, there is one repeated root

The formula used is:

Δ = 4^2 - 4(1)(4) = 16 - 16 = 0

Example 3: Find Vertex

Find the vertex of the parabola y = x² - 4x + 3.

You can use our calculator by entering:

What is the vertex of y = x² - 4x + 3?

The calculator returns: (2, -1)

a = 1, b = -4, c = 3
Vertex x = -b/(2a) = 4/2 = 2
Vertex y = f(2) = 4 - 8 + 3 = -1
Vertex: (2, -1)

The formula used is:

x = ((-(-4)) ÷ 2(1)) = 2, y = -1

Example 4: Factored Form

Convert x² - 5x + 6 = 0 to factored form.

You can use our calculator by entering:

What is the factored form of x² - 5x + 6?

The calculator returns: (x - 2)(x - 3)

Roots: x = 2, x = 3
Factored form: (x - 2)(x - 3)

The formula used is:

x^2 - 5x + 6 = (x - 2)(x - 3)

Example 5: UK Sports Projectile

A football is kicked upward in a match at Wembley Stadium. Find when it hits the ground using h = -5t² + 15t = 0.

You can use our calculator by entering:

When does the ball hit the ground?

The calculator returns: t = 0, t = 3 seconds

-5t² + 15t = 0
t(-5t + 15) = 0
t = 0 or t = 3 seconds

The formula used is:

-5t^2 + 15t = 0 t = 0, t = 3

Example 6: Area Problem

A rectangle has length x and width (10 - x). Find x when area is 24.

You can use our calculator by entering:

What is x when x(10 - x) = 24?

The calculator returns: x = 4 or x = 6

x(10 - x) = 24
10x - x² = 24
x² - 10x + 24 = 0
x = 4 or x = 6

The formula used is:

x^2 - 10x + 24 = 0 x = 4, x = 6

Example 7: Complex Roots

Solve x² + 2x + 5 = 0.

You can use our calculator by entering:

What are the roots of x² + 2x + 5 = 0?

The calculator returns: x = -1 ± 2i

Δ = 2² - 4(1)(5) = 4 - 20 = -16
Since Δ < 0, roots are complex
x = -1 ± 2i

The formula used is:

x = ((-2 ± √(4 - 20)) ÷ 2) = -1 ± 2i

Example 8: UK Business Maximum Profit

A UK business wants to find the price that maximizes profit: P = -2p² + 40p - 100, where p is in pounds.

You can use our calculator by entering:

What price in £ maximizes profit?

The calculator returns: p = £10

Vertex x = -b/(2a) = -40/(2(-2)) = 10
Maximum profit at p = £10

The formula used is:

p = ((-40) ÷ (2(-2))) = 10

Example 9: Perfect Square

Solve x² - 6x + 9 = 0.

You can use our calculator by entering:

What is the root of x² - 6x + 9 = 0?

The calculator returns: x = 3 (repeated)

x² - 6x + 9 = (x - 3)²
x = 3 (repeated root)

The formula used is:

x^2 - 6x + 9 = (x - 3)^2 = 0

Example 10: Factoring

Convert 2x² - 8x + 6 = 0 to factored form.

You can use our calculator by entering:

What is the factored form of 2x² - 8x + 6?

The calculator returns: 2(x - 1)(x - 3)

Roots: x = 1, x = 3
Factored form: 2(x - 1)(x - 3)

The formula used is:

2x^2 - 8x + 6 = 2(x - 1)(x - 3)

Summary

These examples show how our quadratic calculator can handle:

  • •Solving quadratic equations (x² - 5x + 6 = 0)
  • •Calculating discriminants (Δ = b² - 4ac)
  • •Finding vertices of parabolas
  • •Converting to factored form

You can adapt any real-life situation involving quadratic equations to one of these patterns and let the calculator do the work for you.

How to Use

  1. Enter the coefficients a, b, and c of the quadratic equation ax² + bx + c = 0.
  2. Select the type of calculation you need (solve, discriminant, vertex, or factored form).
  3. View the result and step-by-step calculation breakdown.

Frequently asked questions

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a, used to solve quadratic equations of the form ax² + bx + c = 0.

What is the discriminant?

The discriminant (Δ = b² - 4ac) determines the nature of the roots: if Δ > 0, there are two real roots; if Δ = 0, one repeated root; if Δ < 0, two complex roots.

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About this tool

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Check the units, assumptions and rounding before using a result. Health and financial estimates are educational and do not replace individual professional advice.

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