Completing the square - Quadratics - Passport to advanced math

PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Completing the square
Quadratics
Passport to advanced math

LEARNING OBJECTIVE

After this lesson, you will be able to:

· Solve a quadratic equation by completing the square

To answer a question like this:

image Which of the following has the same roots as 30 − 8x = x2y?

A. y = (x − 4)2 − 30

B. y = (x − 4)2 + 30

C. y = (x + 4)2 − 46

D. y = (x + 4)2 + 46

You need to know this:

For quadratics that do not factor easily, you’ll need one of two strategies: completing the square or the quadratic formula (taught in the next lesson). To complete the square, you’ll create an equation in the form (x + h)2 = k, where h and k are constants.

As with factoring, completing the square is most convenient when the coefficient in front of the x2 term is 1.

You need to do this:

Here are the steps for completing the square, demonstrated with a simple example.

Step

Scratchwork

Starting point:

x2 + 8x − 8 = 0

1. Move the constant to the opposite side.

x2 + 8x = 8

2. Divide b, the x-coefficient, by 2, and square the quotient.

image

3. Add the number from the previous step to both sides of the equation and factor.

x2 + 8x + 16 = 8 + 16

(x + 4)(x + 4) = 24

(x + 4)2 = 24

4. Take the square root of both sides.

image

5. Split the result into two equations and solve each one.

image

Explanation:

First, write the equation in standard form: y = x2 + 8x − 30. Move the 30 to the other side to temporarily get it out of the way. Then, complete the square on the right-hand side, by finding image and adding the result to both sides of the equation:

image

The answer choices are all written in factored form. The right side of the equation is a classic quadratic that factors as follows:

image

Finally, solve for y to get y = (x + 4)2 − 46, which makes (C) correct.

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

11. Which of the following equations has the same solutions as x2 + 6x +17 = y?

A. y = (x − 3)2 − 26

B.y = (x − 3)2 + 8

C.y = (x + 3)2 + 8

D. y = (x + 3)2 + 17

12. Which of the following are roots for x2 + 10x − 8 = 0?

A. image

B.image

C.image

D. image

13. Which of the following is equivalent to x2 + 4x + 16 = 0?

A. image

B.image

C.image

D. image

HINT: What can you divide each term by to make the equation in Q14 easier to work with?

14. What value of x satisfies the equation image?

A. image

B.image

C.3

D. image