Answer:
- 8) 4 + 2q²/p² - 4r/p + r²/p²
- 9) (3/4, -9/4)
- 10) (3/8, 41/16)
Step-by-step explanation:
8. ============
Given
- α and β are roots of px² + qx + r = 0
The sum of the roots is α + β = -q/p, the product of then roots αβ = r/p
- (2 + α²)(2 + β²) =
- 4 + 2(α² + β²) + (αβ)² =
- 4 + 2((α + β)² -2αβ) + (αβ)² =
- 4 + 2((-q/p)² - 2r/p) + (r/p)² =
- 4 + 2q²/p² - 4r/p + r²/p²
------------------------------
9. ============
<u>Given function</u>
The minimum point is reached at vertex
<u>The vertex is:</u>
- x = -b/2a
- x = -(-3)/2*2 = 3/4
<u>The corresponding y-coordinate is:</u>
- y = 2(3/4)² - 3(3/4) - 1 = 9/8 - 9/4 - 1 = 1/8(9 - 18 - 9) = - 18/8 = - 9/4
<u>So the point is: </u>
---------------
10. ============
<u>Given function</u>
The maximum is reached at vertex
<u>The vertex is:</u>
- x = -b/2a
- x = -(-3)/2(-4) = -3/8
<u>The corresponding y-coordinate is:</u>
- y = 2 - 3(-3/8) -4(-3/8)² = 2 + 9/8 - 9/16 = 1/16(32 + 18 - 9) = 41/16
<u>So the maximum point is:</u>
Number of dogs in the shelter are 12
<u>Step-by-step explanation:</u>
Step 1:
Given that total animals = 108 and number of ferrets = 1/3 of 108. Find number of ferrets in the shelter.
Number of ferrets = 1/3 of 108 = 1/3 × 108 = 36
Step 2:
Given that of the remaining animals, number of cats = 5/6. Find the remaining animals and the number of cats.
Remaining number of animals = 108 - 36 = 72
Number of cats = 5/6 of 72 = 5/6 × 72 = 60
Step 3:
Find number of dogs in the shelter.
Number of dogs = 108 - (36 + 60) = 108 - 96 = 12
Factor each of the following differences of two squares and write your answer together with solution.
<h3><u>1. x² - 36</u></h3>
Rewrite 
. The difference of squares can be factored using the rule:
.
__________________
<h3><u>2. 49 - x²</u></h3>
Rewrite 49-x² as 7²-x². The difference of squares can be factored using the rule: 
.
Reorder the terms.
__________________
<h3><u>3. 81 - c²</u></h3>
Rewrite 81-c²as 9²-c². The difference of squares can be factored using the rule: .
Reorder the terms.
__________________
<h3><u>4</u><u>.</u><u> </u><u>m²</u><u>n</u><u>²</u><u> </u><u>-</u><u> </u><u>1</u></h3>
Rewrite m²n² - 1 as 
. The difference of squares can be factored using the rule: .
Answer:
2x^2 +2x-4
——————
2x^2-4x+2
Factor out 2 from the expression
2(x^2+x-2)
—————-
2(x^2-2x+1)
Write x as a difference
2(x^2x-x-2)
—————-
2(x^2-2x+1)
Use a^2-2ab+b^2=(ab)^2
2(x^2x-x-2)
—————-
2(x-1)^2
Reduce the fraction with 2
x^2x-x-2
—————-
(x-1)^2
Factor out x from the expression
X*(x^2)-x-2
—————-
(x-1)^2
Factor out negative sign from the expression
X*(x+2)-(x-2)
—————-
(x-1)^2
Factor out x+2 from the expression
(x+2)(x-1)
—————-
(x-1)^2
Simplify the expression
x+2
——
x-1
