1. Practice Quiz Quadratics and Rationals
Monday, April 08, 2019

2. x2 – 4x – 12 (x + 2)(x – 6) One factor of x2 – 4x – 12 is –12 2 –6 –4
Factor using
Diamond method
A. x + 6
B. x – 4
x2 – 4x – 12
–12
C. x + 4 2 –6
(x + 2)(x – 6)
D. x – 6 –4
E. x – 2
Two numbers with
Product of –12 and
Sum of –4

3. If x2 – 3x – 2 = 0, then what is the value of 2x2 – 6x – 11?
Note: 2x2 – 6x – 11
Rewrite
x2 – 3x – 2 = 0
= 2x2 – 6x – 11
Multiply by 2
2x2 – 6x – 4 = 0
Substitute
= 4 – 11 +4 +4
= –7
2x2 – 6x = 4

4. (x + y)(x – y) = x2 – y2 = x2 – y2 = x2 – y2 4 = x2 – y2 x + y = 12 3
If x + y = 12 and , what is the value of x2 – y2 ? 3
x + y = 12
(x + y)(x – y) = x2 – y2
= x2 – y2
= x2 – y2
= x2 – y2

5. (x – y)2 = (x – y)(x – y) = x2 – xy – xy + y2 = x2 – 2xy + y2
If x2 + y2 = 37 and xy = 24, what is the value of (x – y)2? 4
(x – y)2
= (x – y)(x – y)
= x2 – xy – xy + y2
x2 + y2 = 37
= x2 – 2xy + y2
= x2 + y2 – 2xy
= – 2xy
xy = 24
= – 2(24)
= – 48
= –11

6. If (–8x + 3)(–4x2 + 4x + 6) = ax3 + bx2 + cx + d for all real values of x, what is the value of c ?5
(–8x + 3)(–4x2 + 4x + 6)
–4x2 4x 6
–8x
32x3
–32x2
–48x
c = –36 3
–12x2
12x 18
32x3 – 44x2 – 36x + 18
(Add matching colors)

7. x2 + 8x – 20 (x + 10)(x – 2) One factor of x2 + 8x – 20 is –20 10 –2 86
Factor using
Diamond method
A. x + 5
B. x – 10
x2 + 8x – 20
–20
C. x + 4 10 –2
(x + 10)(x – 2)
D. x – 2 8
E. x – 4
Two numbers with
Product of –20 and
Sum of 8

8. 7
If (y – 4)2 = 36 and y < 0, what is the value of y ?
Method 1
Method 2
(y – 4)2 = 36
(y – 4)2 = 36
(y – 4)(y – 4) = 36
y2 – 4y – 4y +16 = 36
y2 – 8y + 16 = 36
y – 4 = 6
–36
–36
y2 – 8y – 20 = 0
y – 4 = 6
y – 4 = –6
(y + 2)(y – 10) = 0 +4 +4 +4 +4
y + 2 = 0
, y – 10 = 0
y = 10
y = –2
y = –2
y = 10

9. x2 – 6x + 8 = 0 (x – 2)(x – 4) = 0 , x – 4 = 0 x – 2 = 0 x = 2 x = 4
If x2 – 6x + 8 = 0 and y = x + 3, then what are the possible values of y ?
x2 – 6x + 8 = 0
(x – 2)(x – 4) = 0
, x – 4 = 0
x – 2 = 0
x = 2
x = 4
Let x = 2:
y = x + 3
Let x = 4:
y = x + 3
= 2 + 3
= 4 + 3
= 5
= 7

10. 9
(2x + 3y)2 – (2x – 3y)2
(2x + 3y)(2x + 3y) – (2x – 3y) (2x – 3y)
(4x2 + 6xy + 6xy + 9y2) – (4x2 – 6xy – 6xy + 9y2)
(4x2 + 12xy + 9y2) – (4x2 – 12xy + 9y2)
4x2 + 12xy + 9y2 – 4x2 + 12xy – 9y2
24xy

11. (x + y)(x – y) = x2 – y2 = x2 – y2 = x2 – y2 10
If and , what is the value of x2 – y2 ? 10
(x + y)(x – y) = x2 – y2
= x2 – y2
= x2 – y2

12. 11
Step 1
Step 2
Step 3

13. 12
If , then c2 =
a2  c = 1  b
a2c = b
a2c b = a2 a2 b
c = a2

14. If v, w, x, y, and z are consecutive integers whose sum is 0, then what
is the value of ? 13
v w x y z
–2 –
Sum = 0
= 0

15. 14
If r and s are positive integers and , then what is the value of s in terms of r?

16. If x and y are positive integers, then which of the following must be equal to ?15

17. x(x – 4) = 1(x – 6) x2 – 4x = x – 6 x2 – 5x + 6 = 0 (x – 2)(x – 3) = 0
Solve for x. 16
x(x – 4) = 1(x – 6)
x2 – 4x = x – 6
–x + 6
–x + 6
x2 – 5x + 6 = 0
(x – 2)(x – 3) = 0
x – 2 = 0 ,
x – 3 = 0
x = 2
x = 3

18. 17 LCD = 4m2 m2 – 8m + 12 = 0 (m – 2)(m – 6) = 0 m2 + 12 = 8m
Solve for m. 17
LCD = 4m2
m2 – 8m + 12 = 0
(m – 2)(m – 6) = 0
m = 8m
m – 2 = 0 ,
m – 6 = 0
m2 – 8m = 0
m = 2
m = 6

19. If , then 18
Reciprocal
= 5 + 5
= 10

20. If and ab  0, then 19
1  a3 = b  x
Reciprocal
a3 = bx

21. B. A. B. 20 Answer If g and m are positive integers, then
which of the following must be equal to ? 20
Answer B.
Another option: Substitute integers for g and m.
Then test each answer.
Substitute
g = 2
m = 3 A. NO B.
YES

22. C. D. E. 20 = 1 If g and m are positive integers, then
which of the following must be equal to ? 20
Another option: Substitute integers for g and m.
Then test each answer.
Substitute
g = 2
m = 3 C.
= 1 NO D. NO E. NO

23. 21
If , then x equals
–6  x = 1  6
–6x = 6
x = –1

24. (y – 5)2 = 0 Find y2 – 2y when y = 5 (y – 5)(y – 5) = 0 (5)2 – 2(5)
If (y – 5)2 = 0, then find the value of y2 – 2y ? 22
(y – 5)2 = 0
Find y2 – 2y
when y = 5
(y – 5)(y – 5) = 0
(5)2 – 2(5)
y – 5 = 0 ,
y – 5 = 0
25 – 10
y = 5
y = 5 15

25. 2x2 5x –3 5x 10x3 25x2 –15x –4 –8x2 –20x 12 23 (2x2 + 5x – 3)(5x – 4)
If (2x2 + 5x – 3)(5x – 4) = ax3 + bx2 + cx + d for all real values of x, what is the value of b ? 23
(2x2 + 5x – 3)(5x – 4)
2x2 5x –3 5x
10x3
25x2
–15x
b = 17 –4
–8x2
–20x 12
10x3 + 17x2 – 35x + 12
(Add matching colors)

26. Solve for x. 24
2(3x) = 1(5x + 9)
6x = 5x + 9
–5x
–5x
x =

27. x6 = xn x6 = xn 6 = n If , then n = 25 Add exponents in numerator
Subtract exponents
x6 = xn
x6 = xn
Exponents are equal
6 = n
Answer