Calculator Use

This calculator uses the "complete the square" method to solve quadratic equations and second degree polynomial equations in the form

ax² + bx + c = 0, where a ≠ 0

The solution shows the work required to solve a quadratic equation for real and complex roots by completing the square.

What is Completing the Square?

Completing the square is a method of solving quadratic equations by changing the left side of the equation so that it is the square of a binomial.

You can use the complete the square method when it is not possible to solve the equation by factoring.

First, make sure that the a term is 1. If it is not 1, divide both sides of the equation by the a term and then continue to complete the square as explained below.

How to Complete the Square

It takes a few steps to complete the square of a quadratic equation.

First, arrange your equation to the form ax² + bx + c = 0
If a ≠ 1, divide both sides of your equation by a. Your b and c terms may be fractions after this step.
Move the c term to the right side of the equation by subtracting it from or adding it to both sides of the equation
Take the b term, divide it by 2, and then square it
Add this result to both sides of the equation
Rewrite the perfect square on the left to the form (x + y)²
Take the square root of both sides
Isolate x on the left by subtracting or adding the numeric constant on both sides
Solve for x. Remember you will have 2 solutions, a positive solution and a negative solution, because you took the square root of the right side of the equation.

Completing the Square when a is Not Equal to 1

To complete the square when a is greater than 1 or less than 1 but not equal to 0, divide both sides of the equation by a. This is the same as factoring out the value of a from all other terms.

As an example let's complete the square for this quadratic equation:

\[ 2x^2 - 12x + 7 = 0 \]

a ≠ 1, and a = 2, so divide all terms by 2

\[ \dfrac{2}{2}x^2 - \dfrac{12}{2}x + \dfrac{7}{2} = \dfrac{0}{2} \]

which gives you

\[ x^2 - 6x + \dfrac{7}{2} = 0 \]

Continue to solve this quadratic equation with the completing the square method described above.

Completing the Square When b is 0

When you do not have an x term because b = 0, the equation is easier to solve. You only need to solve for the x squared term.

As an example let's find the solution by completing the square for

\[ x^2 + 0x - 4 = 0 \]

Eliminate the b term

\[ x^2 - 4 = 0 \]

Keep the x term on the left and move the constant to the right side by adding it to both sides

\[ x^2 = 4\]

Take the square root of both sides

\[ x = \pm \sqrt[]{4} \]

Because you took the square root you will have 2 answers -- a positive solution and a negative solution

\[ x = + 2 \]

\[ x = - 2 \]

References

Visit Deciding Which Method to Use when Solving Quadratic Equations to help determine when to use the "completing the square" method to solve a quadratic equation.