Solving Quadratics by Completing the Square (2.4.2)

Question 1

• Completing the square is a process used to solve quadratic equations. In this method

1. Move constants across the equal sign away from the variable.
2. Divide the coefficient of the unsquared term by two and square that answer.
3. Add that answer to both sides of the equation to create a perfect square trinomial with the variable.
4. Factor the perfect square trinomial.
5. Take the square root of both sides.
6. Add or subtract constants as necessary to solve for the variable.

Answer 1

• Completing the square is a process used to solve quadratic equations. In this method

1. Move constants across the equal sign away from the variable.
2. Divide the coefficient of the unsquared term by two and square that answer.
3. Add that answer to both sides of the equation to create a perfect square trinomial with the variable.
4. Factor the perfect square trinomial.
5. Take the square root of both sides.
6. Add or subtract constants as necessary to solve for the variable.

Question 2

You have two ways to solve this problem.

You can move the 25 to the left side and then factor. Or, as shown here, you can take the square root of both sides of the equation. Either way, you will find that x = 5, –5.

Remember: You get both a positive and a negative root.

A problem like this one invites you to factor it so you can
solve for x.

However, it won’t factor.

Remembering the problem above, look for a way to take the square root here too.

Step one: Move the +1 to the right side.

Step Two: Divide the coefficient of the 6x by 2 and square to get 9.

Step Three: Add 9 to both sides of the equation.

You have created a perfect square on the side with the
variable.

Step Four: Factor your perfect square.

Step Five: Take the square root of both sides of the equation.

Step Six: Subtract +3 to isolate x. This gives you two
answers for x, as expected.

Note: In this specific example, you can simplify the radical
rather than leaving 8 under the radical.

Answer 2

You have two ways to solve this problem.

You can move the 25 to the left side and then factor. Or, as
shown here, you can take the square root of both sides of the equation. Either way, you will find that x = 5, –5.

Remember: You get both a positive and a negative root.

A problem like this one invites you to factor it so you can
solve for x.

However, it won’t factor.

Remembering the problem above, look for a way to take the square root here too.

Step one: Move the +1 to the right side.

Step Two: Divide the coefficient of the 6x by 2 and square
to get 9.

Step Three: Add 9 to both sides of the equation.
You have created a perfect square on the side with the
variable.

Step Four: Factor your perfect square.

Step Five: Take the square root of both sides of the equation.

Step Six: Subtract +3 to isolate x. This gives you two
answers for x, as expected.

Note: In this specific example, you can simplify the radical
rather than leaving 8 under the radical.