Solve for Equal Roots Calculator

A quadratic equation with equal roots is one where the two solutions are identical. This occurs when the discriminant of the quadratic equation is zero. The calculator on this page helps you find such equations by solving for the coefficients when given the equal root value.

What are equal roots in quadratic equations?

In a quadratic equation of the form ax² + bx + c = 0, the roots are the values of x that satisfy the equation. When the equation has equal roots, both roots are the same value. This happens when the quadratic equation has a perfect square as its factor.

For example, the equation x² - 6x + 9 = 0 has roots at x = 3 and x = 3, meaning it has equal roots.

Equal roots occur when the discriminant (b² - 4ac) equals zero. This means the quadratic equation touches the x-axis at exactly one point.

How to solve for equal roots

To find a quadratic equation with equal roots, you need to determine the coefficients a, b, and c such that the discriminant is zero. Here's the step-by-step process:

Start with the general quadratic equation: ax² + bx + c = 0
Calculate the discriminant: D = b² - 4ac
Set the discriminant equal to zero: b² - 4ac = 0
Solve for one of the coefficients (a, b, or c) in terms of the others

This process allows you to find the relationship between the coefficients when the quadratic equation has equal roots.

The formula for equal roots

The key relationship for equal roots is derived from the discriminant condition:

For a quadratic equation ax² + bx + c = 0 to have equal roots, the discriminant must satisfy:

b² - 4ac = 0

This can be rearranged to solve for any one coefficient in terms of the others.

For example, solving for c gives:

c = (b²)/(4a)

Worked example

Let's find a quadratic equation with equal roots at x = 5.

1. Assume the equation is in the form x² - bx + c = 0 (a = 1)

2. The roots are x = 5 and x = 5, so the sum of roots is 10 = b

3. The product of roots is 25 = c

4. Therefore, the equation is x² - 10x + 25 = 0

This equation has equal roots at x = 5, as the discriminant is 100 - 100 = 0.

Frequently Asked Questions

What does it mean for a quadratic equation to have equal roots?: It means the quadratic equation touches the x-axis at exactly one point, and both solutions are identical.
How do you know if a quadratic equation has equal roots?: Check if the discriminant (b² - 4ac) equals zero. If it does, the equation has equal roots.
Can a quadratic equation have equal roots and be a perfect square?: Yes, when the discriminant is zero, the quadratic can be written as a perfect square of a linear term.
What's the difference between equal roots and no real roots?: Equal roots occur when the discriminant is zero, while no real roots occur when the discriminant is negative.
How can I use the equal roots calculator to find a quadratic equation?: Enter the value of the equal root and the calculator will show you the relationship between the coefficients that satisfy the equal roots condition.