1. In this session you will learn to In this lesson you will learn to
•Graph quadratic functions, •Solve quadratic equations by
•Solve quadratic equations. finding the square root.
•Graph exponential functions •Solve quadratic equations by
•Solve problems involving completing the square.
exponential growth and decay.
•Recognize and extend geometric
sequences.
The Gateway Arch in January 2008
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Click to continue.

2. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,

3. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6

4. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6
then take the square root of both sides. 3x 2 2 6

5. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6
then take the square root of both sides. 3x 2 2 6
Simplify. 3x 2 6

6. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6
then take the square root of both sides. 3x 2 2 6
Simplify. 3x 2 6
Solve for the unknown variable by 3x 2 6
adding 2

7. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6
then take the square root of both sides. 3x 2 2 6
Simplify. 3x 2 6
Solve for the unknown variable by 3x 2 6
adding 2
and dividing by 3.
2 6
x 3

8. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.

9. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12 x ___ 1

10. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12 x ___ 1
•Complete the square by adding the square of
x 2
12x 12 2 1 12 2
half the coefficient of the second term. 2 2

11. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12 x ___ 1
•Complete the square by adding the square of
x 2
12x 12 2 1 12 2
half the coefficient of the second term. 2 2
•Simplify.
x 2 12 x 36 35

12. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12 x ___ 1
•Complete the square by adding the square of
x 2
12x 12 2 1 12 2
half the coefficient of the second term. 2 2
•Simplify.
x 2 12 x 36 35
•The trinomial on the left is now a perfect square.
Factor the trinomial on the left.
x 62 35

13. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12 x ___ 1
•Complete the square by adding the square of
x 2
12x 12 2 1 12 2
half the coefficient of the second term. 2 2
•Simplify.
x 2 12 x 36 35
•The trinomial on the left is now a perfect square.
Factor the trinomial on the left.
x 62 35
•Take the square root of both sides. x 6 35

14. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve x 2 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12 x ___ 1
•Complete the square by adding the square of
x 2
12x 12 2 1 12 2
half the coefficient of the second term. 2 2
•Simplify.
x 2 12 x 36 35
•The trinomial on the left is now a perfect square.
Factor the trinomial on the left.
x 62 35
•Take the square root of both sides. x 6 35
•Solve for the unknown variable by adding or
subtracting.
x 6 35

15. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.

16. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3

17. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3
•Subtract 7 . x2 6x _ 3 7
2
x2 6x _ 1
2
2

18. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3
•Subtract 7 . x2 6x _ 3 7
2
x2 6x _ 1
2
2
•Complete the square by adding the square of
x 2
6x 62 1 62
half the coefficient of the second term. 2 2 2
•Simplify. x2 6x 32 1
2
32 x 2 6 x 9 17
2

19. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3
•Subtract 7 . x2 6x _ 3 7
2
x2 6x _ 1
2
2
•Complete the square by adding the square of
x 2
6x 62 1 62
half the coefficient of the second term. 2 2 2
•Simplify. x2 6x 32 1
2
32 x 2 6 x 9 17
2
•The trinomial on the left is now a perfect square. x 32 17
2
Factor the trinomial on the left.

20. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3
•Subtract 7 . x2 6x _ 3 7
2
x2 6x _ 1
2
2
•Complete the square by adding the square of
x 2
6x 62 1 62
half the coefficient of the second term. 2 2 2
•Simplify. x2 6x 32 1
2
32 x 2 6 x 9 17
2
•The trinomial on the left is now a perfect square. x 32 17
2
Factor the trinomial on the left.
•Take the square root of both sides. x 3 17
2

21. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3
•Subtract 7 . x2 6x _ 3 7
2
x2 6x _ 1
2
2
•Complete the square by adding the square of
x 2
6x 62 1 62
half the coefficient of the second term. 2 2 2
•Simplify. x2 6x 32 1
2
32 x 2 6 x 9 17
2
•The trinomial on the left is now a perfect square. x 32 17
2
Factor the trinomial on the left.
•Take the square root of both sides. x 3 17
2
•Solve for the unknown variable by adding or x 3 17
subtracting. 2

22. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.

23. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0

24. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2

25. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.

26. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.
2 2 2
•Simplify. x 5
2 x 5
4
7
2
5
4 x2 5
2 x 25
16
7
2
25
16
81
16

27. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.
2 2 2
•Simplify. x 5
2 x 5
4
7
2
5
4 x2 5
2 x 25
16
7
2
25
16
81
16
•The trinomial on the left is now a perfect square. x 5
2
81
4 16
Factor the trinomial on the left.

28. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.
2 2 2
•Simplify. x 5
2 x 5
4
7
2
5
4 x2 5
2 x 25
16
7
2
25
16
81
16
•The trinomial on the left is now a perfect square. x 5
2
81
4 16
Factor the trinomial on the left.
•Take the square root of both sides. x 5
4
81
4
9
2

29. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.
2 2 2
•Simplify. x 5
2 x 5
4
7
2
5
4 x2 5
2 x 25
16
7
2
25
16
81
16
•The trinomial on the left is now a perfect square. x 5
2
81
4 16
Factor the trinomial on the left.
•Take the square root of both sides. x 5
4
81
4
9
2
•Solve for the unknown variable by adding or x 5
4
9
4
14
4 , 4
4
subtracting.

30. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.
2 2 2
•Simplify. x 5
2 x 5
4
7
2
5
4 x2 5
2 x 25
16
7
2
25
16
81
16
•The trinomial on the left is now a perfect square. x 5
2
81
4 16
Factor the trinomial on the left.
•Take the square root of both sides. x 5
4
81
4
9
2
•Solve for the unknown variable by adding or x 5
4
9
4
14
4 , 4
4
subtracting.
Reduce: x 7
2 , 1