Download The graphs coincide. Therefore, the trinomial has been factored

The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors are (y – 8)(y – 9).
8-6 Solving x^2 + bx + c = 0
Factor each polynomial. Confirm your answers using a graphing
calculator.
2
13. y − 17y + 72
SOLUTION: In this trinomial, b = −17 and c = 72, so m + p is negative and mp is
positive. Therefore, m and p must both be negative. List the factors of 72
and identify the factors with a sum of −17.
Factors of 72
Sum
−73
−1, −72
−38
−2, −36
−27
−3, −24
−22
−4, −18
−18
−6, −12
−17
−8, −9
The correct factors are −8 and −9.
2
15. n − 2n − 35
SOLUTION: In this trinomial, b = −2 and c = −35, so m + p is negative and mp is
negative. Therefore, m and p must have opposite signs. List the factors
of −35 and identify the factors with a sum of −2.
Sum
Factors of −35
34
−1, 35
2
−5, 7
−7, 5
−2
−35, 1
−34
The correct factors are −7 and 5.
2
Confirm by graphing Y1 = n – 2n – 35 and Y2 = (n – 7)(n + 5) on the
same screen.
2
Confirm by graphing Y1 = y − 17y + 72 and Y2 = (y – 8)(y – 9) on the
same screen.
The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors are (n – 7)(n + 5).
The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors are (y – 8)(y – 9).
2
15. n − 2n − 35
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SOLUTION: In this trinomial, b = −2 and c = −35, so m + p is negative and mp is
17. 40 − 22x + x
2
SOLUTION: 2
First rearrange the polynomial in decreasing order, x – 22x + 40.
In this trinomial, b = −22 and c = 40, so m + p is negative and mp is
positive. Therefore, m and p must both be negative. List the factors of 40
Page 1
and identify the factors with a sum of −22.
Factors of 40
Sum
−41
−1, −40
The graphs coincide. Therefore, the trinomial has been factored
correctly.x^2
√ + bx + c = 0
8-6 Solving
The factors are (n – 7)(n + 5).
17. 40 − 22x + x
2
The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors are (x – 2)(x – 20).
2
19. −42 − m + m
SOLUTION: SOLUTION: 2
First rearrange the polynomial in decreasing order, x – 22x + 40.
In this trinomial, b = −22 and c = 40, so m + p is negative and mp is
positive. Therefore, m and p must both be negative. List the factors of 40
and identify the factors with a sum of −22.
Factors of 40
Sum
−41
−1, −40
−22
−2, −20
−14
−4, −10
−13
−5, −8
The correct factors are −2 and −20.
2
Confirm by graphing Y1 = 40 – 22x + x and Y2 = (x – 2)(x – 20) on
the same screen.
2
First rearrange the polynomial in decreasing order m – m – 42.
In this trinomial, b = –1 and c = –42, so m + p is negative and mp is
negative. Therefore, m and p must have opposite signs. List the factors
of –42 and identify the factors with a sum of –1.
Factors of –42
Sum
41
–1, 42
19
–2, 21
11
–3, 14
1
–6, 7
–7, 6
–1
–14, 3
–11
–21, 2
–19
–42, 1
–41
The correct factors are –7 and 6.
2
Confirm by graphing Y1 = –42 – m + m and Y2 = (m + 6)(m – 7) on the
same screen.
The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors are (x – 2)(x – 20).
2
19. −42 − m + m
The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors are (m + 6)(m – 7).
SOLUTION: 2
First rearrange the polynomial in decreasing order m – m – 42.
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In this trinomial, b = –1 and c = –42, so m + p is negative and mp is
negative. Therefore, m and p must have opposite signs. List the factors
of –42 and identify the factors with a sum of –1.
Solve each equation. Check your solutions.
2
21. y + y = 20
SOLUTION: Page 2
The graphs coincide. Therefore, the trinomial has been factored
correctly. √
The factors
(m ++ c6)(m
8-6 Solving
x^2are
+ bx
= 0 – 7).
Solve each equation. Check your solutions.
and
2
21. y + y = 20
SOLUTION: Rewrite the equation with 0 on the right side.
The solutions are –5 and 4.
List the factors of –20 and identify the factors with a sum of 1.
Factors of –20
Sum
1, –20
–19
–1, 20
19
–8
2,– 10
8
– 2, 10
–1
4, – 5
1
– 4, 5
2
23. a + 11a = −18
SOLUTION: Rewrite the equation with 0 on the right side.
List the factors of 18 and identify the factors with a sum of 11.
Factors of 18
Sum
1, 18
19
2, 9
11
3, 6
19
The roots are –5 and 4. Check by substituting –5 and 4 in for y in the
original equation.
The roots are –2 and –9. Check by substituting –2 and –9 in for a in the
original equation.
and
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The solutions are –5 and 4.
and
Page 3
8-6 Solving x^2 + bx + c = 0
The roots are –2 and –9. Check by substituting –2 and –9 in for a in the
original equation.
The solutions are –2 and –9.
2
25. x − 18x = −32
SOLUTION: Rewrite the equation with 0 on the right side.
and
The solutions are –2 and –9.
List the factors of 32 and identify the factors with a sum of –18.
Factors of 32
Sum
–1, –32
–33
–2, –16
–18
–4, –8
–12
2
25. x − 18x = −32
SOLUTION: Rewrite the equation with 0 on the right side.
The roots are 16 and 2. Check by substituting 16 and 2 in for x in the
original equation.
List the factors of 32 and identify the factors with a sum of –18.
Factors of 32
Sum
–1, –32
–33
–2, –16
–18
–4, –8
–12
The roots are 16 and 2. Check by substituting 16 and 2 in for x in the
original equation.
and
The solutions are 2 and 16.
2
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27. d + 56 = −18d
SOLUTION: Rewrite the equation with 0 on the right side.
Page 4
8-6 Solving x^2 + bx + c = 0
The solutions are 2 and 16.
The solutions are −4 and −14.
2
27. d + 56 = −18d
SOLUTION: Rewrite the equation with 0 on the right side.
List the factors of 56 and identify the factors with a sum of 18.
Factors of 56
Sum
1, 56
57
2, 38
40
4, 14
18
7, 8
15
The roots are −4 and −14. Check by substituting −4 and −14 in for d in
the original equation.
2
29. h + 48 = 16h
SOLUTION: Rewrite the equation with 0 on the right side.
List the factors of 48 and identify the factors with a sum of –16.
Factors of 48
Sum
–1, –48
–49
–2, –24
–26
–3, –16
–19
–4, –12
–16
–14
6,
8
– –
The roots are 4 and 12. Check by substituting 4 and 12 in for h in the
original equation.
and
and
The solutions are −4 and −14.
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2
29. h + 48 = 16h
SOLUTION: Page 5
The solutions are 4 and 12.
2
31. GEOMETRY A rectangle has an area represented by x − 4x − 12
−3, 4
1
Then area of the rectangle is (x + 2)(x – 6). Area is found by multiplying
the length by the width. Because the length is x + 2, the width must be x
− 6.
8-6 Solving x^2 + bx + c = 0
and
CCSS STRUCTURE Factor each polynomial.
2
2
33. q + 11qr + 18r
SOLUTION: In this trinomial, b = 11 and c = 18, so m + p is positive and mp is
positive. Therefore, m and p must both be positive. List the factors of 18
and identify the factors with a sum of 11.
The solutions are 4 and 12.
2
31. GEOMETRY A rectangle has an area represented by x − 4x − 12
square feet. If the length is x + 2 feet, what is the width of the rectangle?
Factors of 18
1, 18
2, 9
3, 6
SOLUTION: 2
The area of the rectangle is x − 4x − 12.
List the factors of −12 and identify the factors with a sum of −4.
Factors of −12
Sum
1, −12
−11
−1, 12
11
2, −6
−4
−2, 6
4
−1
3, −4
1
−3, 4
Then area of the rectangle is (x + 2)(x – 6). Area is found by multiplying
the length by the width. Because the length is x + 2, the width must be x
− 6.
CCSS STRUCTURE Factor each polynomial.
2
2
33. q + 11qr + 18r
SOLUTION: In this trinomial, b = 11 and c = 18, so m + p is positive and mp is
positive. Therefore, m and p must both be positive. List the factors of 18
and identify the factors with a sum of 11.
Sum 19
11
9
The correct factors are 2 and 9.
The trinomial has two variables q and r. The first term in each binomial
will have q's, the second term will have the r along with the factors. 2
35. x − 6xy + 5y
2
SOLUTION: In this trinomial, b = −6 and c = 5, so m + p is negative and mp is positive.
Therefore, m and p must both be negative. List the negative factors of 5
and identify the factors with a sum of −6.
Factors of 5
Sum
−6
−1, −5
The correct factors are −1 and −6. The trinomial has two variables x and y . The first term in each binomial
will have x's, the second term will have the y along with the factors. Factors of 18
1, 18
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2, 9
3, 6
Sum 19
11
9
37. SWIMMING The length of a rectangular swimming pool is 20 feetPage 6
greater than its width. The area of the pool is 525 square feet.
a. Define a variable and write an equation for the area of the pool.
8-6 Solving x^2 + bx + c = 0
2
35. x − 6xy + 5y
2
SOLUTION: In this trinomial, b = −6 and c = 5, so m + p is negative and mp is positive.
Therefore, m and p must both be negative. List the negative factors of 5
and identify the factors with a sum of −6.
Factors of 5
Sum
−6
−1, −5
The correct factors are −1 and −6. The trinomial has two variables x and y . The first term in each binomial
will have x's, the second term will have the y along with the factors. 37. SWIMMING The length of a rectangular swimming pool is 20 feet
greater than its width. The area of the pool is 525 square feet.
a. Define a variable and write an equation for the area of the pool.
b. Solve the equation.
c. Interpret the solutions. Do both solutions make sense? Explain.
SOLUTION: a. Sample answer: Let w = width. Then the length is 20 feet greater than
its width, so = w + 20. The area, which is 525 square feet, is found by multiplying the length times the width. Therefore the area is (w + 20)w =
525.
b. List the factors of −525 and identify the factors with a sum of 20.
Factors of −525
Sum
1, −525
−524
524
−1, 525
5, −105
−100
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100
−5, 105
−68
7, −75
68
−7, 75
List the factors of −525 and identify the factors with a sum of 20.
Factors of −525
Sum
1, −525
−524
524
−1, 525
5, −105
−100
100
−5, 105
−68
7, −75
68
−7, 75
−20
15, −35
20
−15, 35
−4
21, −25
4
−21, 25
The width cannot be negative, therefore it is 15 feet. The length is 15 +
20, 35 feet.
c. The solution of 15 means that the width is 15 ft. The solution −35 does
not make sense because the width cannot be negative.
GEOMETRY Find an expression for the perimeter of a rectangle
with the given area.
2
39. A = x + 13x − 90
SOLUTION: 2
The area of the rectangle is x + 13x − 90.
In this trinomial, b = 13 and c = − 90, so m + p is positive and mp is
positive. Therefore, m and p must both be positive. List the factors of −
90 and identify the factors with a sum of 13.
Sum
Factors of − 90
1, −90
−89
−1, 90
89
2, −45
−43
−2, 45
43
−27
3, −30
27
−3, 30
Page 7
−13
5, −18
13
−5, 18
The width cannot be negative, therefore it is 15 feet. The length is 15 +
20, 35 feet.
c. The solution
of 15
8-6 Solving
x^2 + bx
+ cmeans
= 0 that the width is 15 ft. The solution −35 does
not make sense because the width cannot be negative.
GEOMETRY Find an expression for the perimeter of a rectangle
with the given area.
2
39. A = x + 13x − 90
So, an expression for the perimeter of the rectangle is 4x + 26.
2
41. ERROR ANALYSIS Jerome and Charles have factored x + 6x − 16.
Is either of them correct? Explain your reasoning.
SOLUTION: 2
The area of the rectangle is x + 13x − 90.
In this trinomial, b = 13 and c = − 90, so m + p is positive and mp is
positive. Therefore, m and p must both be positive. List the factors of −
90 and identify the factors with a sum of 13.
Sum
Factors of − 90
1, −90
−89
−1, 90
89
2, −45
−43
−2, 45
43
−27
3, −30
3,
30
27
−
−13
5, −18
13
−5, 18
−9
6, −15
6,
15
9
−
−1
9, −10
1
−9, 10
SOLUTION: Charles is correct. In this trinomial, b = 6 and c = − 16, so m + p is positive and mp is
positive. Therefore, m and p must both be positive. List the factors of −
16 and identify the factors with a sum of − 6.
Sum
Factors of − 16
1, −16
−15
−1, 16
15
2, −8
−6
−2, 8
6
4, −4
0
2
The correct factors are −2, 8. Jerome’s answer once multiplied is x − 6x
− 16. The middle term should be positive.
2
The area of the rectangle x + 13x − 90 factors to (x + 18)(x – 5). Area is found by multiplying the length by the width, so the length is x +
18 and the width is x − 5.
So, an expression for the perimeter of the rectangle is 4x + 26.
Jerome
41. ERROR ANALYSIS
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2
and Charles have factored x + 6x − 16.
Is either of them correct? Explain your reasoning.
Page 8