section 7.2 solving quadratic equations by completing the square
DESCRIPTION
A key to working the problems in this section is to recognize perfect square trinomials. Recall that: Square of a sum: Square of a difference:TRANSCRIPT
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.