Section 7.2

Solving Quadratic Equations by Completing the Square

7.2 Lecture Guide: Solving Quadratic Equations by Completing the Square

Objective 1: Determine the constant term in a perfect square trinomial.

22 22 __________A AB B

22 22 __________A AB B

A key to working the problems in this section is to recognizeperfect square trinomials. Recall that:

Square of a sum:

Square of a difference:

Write each equation so the left side is expressed as the square of a binomial.

1. 2 6 9 5x x

Write each equation so the left side is expressed as the square of a binomial.

2. 2 8 16 81x x

Fill in the missing constant term that is needed to make each expression a perfect square trinomial.

3. 2 10 _____x x

Fill in the missing constant term that is needed to make each expression a perfect square trinomial.

4. 2 16 _____x x

Objective 2: Solve quadratic equations by completing the square.Rewrite the left side of each equation so that it is a perfect square, and then solve this equation by using extraction of roots.

5. 2 12 0x x

Objective 2: Solve quadratic equations by completing the square.Rewrite the left side of each equation so that it is a perfect square, and then solve this equation by using extraction of roots.

6. 2 10 9x x

Completing the SquareStep 1. Write the equation with

the _______________ term on the right side.

Example:

Step 2. Divide both sides of the equation by the coefficient of

to obtain a coefficient of ______ for .

Step 3. Take one-half of the coefficient of x, square this number, and add the result to _______________ sides of the equation.

2 6 12 0x x

2x2x

Step 4. Write the left side of the equation as a perfect _______________.

Example:

Step 5. Solve this equation by extraction of _______________.

Completing the Square

2 6 12 0x x

Solve the following equations using the method of completing the square.

7. 2 2 4 7x x

Solve the following equations using the method of completing the square.

8. 2 8 5 0x x

Solve the following equations using the method of completing the square.

9. 23 9 3x x

Solve the following equations using the method of completing the square.

10. 22 20 4 1x x

Solve the following equations using the method of completing the square.

11. 2 7 52 2

x x

Solve the following equations using the method of completing the square.

12. 14 40x x

13. Use the graph of to: 2 5y x

(a) Solve

(b) Solve

(c) Solve

2 5y x 2 5 0x 2 5 0x 2 5 0x

-6 6

-6

6y

x

55

2 5 0.x

2 5 0.x

2 5 0.x

14. Construct a quadratic equation in x that has the given solutions.

(a) and 34

x 34

x

14. Construct a quadratic equation in x that has the given solutions.

(b) and

34

x 3

4x

15. A square picture is surrounded by a mat and then framed. The width of the square mat is 1.5 times the width of the picture. If the area covered by the mat is 145 , determine the width of the picture. Round to the nearest tenth of an inch.

2in

(a) Identify the variable: Let w = the _______________ of the picture in inches. Let _________ = the ______________ of the mat in

inches.

(b) Write the word equation: Area covered by mat = Total area covered by mat and picture ________________________

(c) Translate the word equation into an algebraic equation: ____________ = _______________ - ________________

(d) Solve this equation:

15. A square picture is surrounded by a mat and then framed. The width of the square mat is 1.5 times the width of the picture. If the area covered by the mat is 145 , determine the width of the picture. Round to the nearest tenth of an inch.

2in

(e) Write a sentence that answers the question:

(f) Is this answer reasonable?

2in

15. A square picture is surrounded by a mat and then framed. The width of the square mat is 1.5 times the width of the picture. If the area covered by the mat is 145 , determine the width of the picture. Round to the nearest tenth of an inch.