a rectangle has an area of $x^3 - 5x^2 - 3x + 15$, if the length is $x^2 - 3$, what is the width?
a. $x + 5$
b. $x - 5$
c. $x + 3$
d. $x + 15$
which of the following is equivalent to $\frac{2x^3 - 4x^2 - 16x - 70}{2x - 10}$
f $x(x + 3)$
g $2x^3 - 4x^2 - 8x - 7$
h $x^2 + 3x + 7$
j $2x^2 - 8x - 35$
if a and b are the solutions to $0 = x^2 - 9$, what is the value of $a + b$
select the expression that is equivalent to the expression below.
$x^2 + 6x + 4$
f. $(x + 3)^2 + 5$
g. $(x + 3)^2 - 5$
h. $(x - 3)^2 + 5$
j. $(x - 3)^2 - 5$
what are the solutions to $3x^2 + 12x + 6 = 0$
$sqrt{k + 2} - x = 0$
in the equation above, k is a constant. if $x = 9$, what is the value of k
given the function of $f(x) = 3(x - a)^2 + 4$ is translated to the right, what could be a possible value of a
if $x^4 + 2x^3 + 5x - 1$ and $3x^3 - x^2 + 3$ were combined to $x^4 + cx^3 - x^2 + 5x + 2$, what would be the value of c

Question

Accepted Answer

### First Question (Rectangle Width)
# Explanation:
## Step1: Recall Area Formula
The area of a rectangle is $ \text{Area} = \text{length} \times \text{width} $, so $ \text{width} = \frac{\text{Area}}{\text{length}} $.
## Step2: Perform Polynomial Division
We need to divide $ x^3 - 5x^2 - 3x + 15 $ by $ x^2 - 3 $. Using polynomial long division or factoring:
Factor the numerator: $ x^3 - 5x^2 - 3x + 15 = x^2(x - 5) - 3(x - 5) = (x^2 - 3)(x - 5) $.
Then $ \frac{(x^2 - 3)(x - 5)}{x^2 - 3} = x - 5 $ (canceling $ x^2 - 3 $ as long as $ x^2
eq 3 $).
# Answer: B. $ X - 5 $

### Second Question (Equivalent Rational Expression)
# Explanation:
## Step1: Simplify the Numerator and Denominator
First, factor the numerator and denominator. The denominator $ 2x - 10 = 2(x - 5) $. Let's try to factor the numerator $ 2x^3 - 4x^2 - 16x - 70 $. We can factor out a 2: $ 2(x^3 - 2x^2 - 8x - 35) $. Now perform polynomial division of $ x^3 - 2x^2 - 8x - 35 $ by $ x - 5 $ (since denominator has $ x - 5 $ after factoring out 2).
Using synthetic division with root 5:
Coefficients: 1 | -2 | -8 | -35
Bring down 1. Multiply by 5: 5. Add to -2: 3. Multiply by 5: 15. Add to -8: 7. Multiply by 5: 35. Add to -35: 0. So the quotient is $ x^2 + 3x + 7 $. Then the original expression becomes $ \frac{2(x - 5)(x^2 + 3x + 7)}{2(x - 5)} = x^2 + 3x + 7 $ (canceling 2 and $ x - 5 $ for $ x
eq 5 $).
# Answer: H. $ x^2 + 3x + 7 $

### Third Question (Sum of Solutions)
# Explanation:
## Step1: Solve the Quadratic Equation
The equation is $ x^2 - 9 = 0 $, which factors as $ (x - 3)(x + 3) = 0 $. So the solutions are $ x = 3 $ and $ x = -3 $. Let $ a = 3 $, $ b = -3 $.
## Step2: Calculate $ a + b $
$ a + b = 3 + (-3) = 0 $.
# Answer: 0

### Fourth Question (Equivalent Quadratic Expression)
# Explanation:
## Step1: Recall Completing the Square
For a quadratic $ x^2 + bx + c $, completing the square gives $ (x + \frac{b}{2})^2 + (c - \frac{b^2}{4}) $. Here, $ x^2 + 6x + 4 $, $ b = 6 $, so $ \frac{b}{2} = 3 $. Then $ (x + 3)^2 = x^2 + 6x + 9 $.
## Step2: Adjust to Match the Original Expression
We have $ x^2 + 6x + 4 = (x + 3)^2 - 5 $ (since $ 9 - 5 = 4 $).
# Answer: G. $ (x + 3)^2 - 5 $

### Fifth Question (Solutions to Quadratic Equation)
# Explanation:
## Step1: Use Quadratic Formula
The quadratic is $ 3x^2 + 12x + 6 = 0 $. Divide all terms by 3: $ x^2 + 4x + 2 = 0 $. The quadratic formula is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = 4 $, $ c = 2 $.
## Step2: Calculate the Discriminant and Solutions
Discriminant $ D = 16 - 8 = 8 $. So $ x = \frac{-4 \pm \sqrt{8}}{2} = \frac{-4 \pm 2\sqrt{2}}{2} = -2 \pm \sqrt{2} $.
# Answer: $ x = -2 + \sqrt{2} $ and $ x = -2 - \sqrt{2} $

### Sixth Question (Value of $ k $)
# Explanation:
## Step1: Substitute $ x = 9 $ into the Equation
The equation is $ \sqrt{k + 2} - x = 0 $. Substitute $ x = 9 $: $ \sqrt{k + 2} - 9 = 0 $.
## Step2: Solve for $ k $
Add 9 to both sides: $ \sqrt{k + 2} = 9 $. Square both sides: $ k + 2 = 81 $. Subtract 2: $ k = 79 $.
# Answer: 79

### Seventh Question (Possible Value of $ a $)
# Explanation:
## Step1: Recall Horizontal Translations of Functions
The function $ f(x) = 3(x - a)^2 + 4 $ is a parabola. The vertex form is $ f(x) = a(x - h)^2 + k $, where $ h $ is the x - coordinate of the vertex. A translation to the right means the vertex moves to the right. The original vertex (without translation consideration) is at $ x = a $. If we translate to the right, we need to see the effect on $ a $. Wait, actually, the function $ y = (x - h)^2 $ shifts right by $ h $ units. So if we translate $ f(x) = 3(x - a)^2 + 4 $ to the right, we can rewrite it as $ f(x) = 3(x - (a + c))^2 + 4 $ where $ c>0 $. So $ a $ can be any real number, but if we consider the direction, for example, if the original vertex is at $ x = a $, and we move right, $ a $ could be a number such that when we shift right, the new vertex is at $ x = a + c $, $ c>0 $. But more simply, the value of $ a $ can be any real number, but if we take a possible value, say $ a = 0 $, but actually, the key is that the horizontal shift of $ y = (x - h)^2 $ is determined by $ h $. If we translate to the right, we can have $ a $ as any number, but a possible value could be, for example, $ a = 1 $ (since shifting right from $ x = 1 $ would be $ x = 1 + c $, $ c>0 $). Wait, maybe a better way: the function $ f(x) = 3(x - a)^2 + 4 $ has vertex at $ (a, 4) $. Translating to the right means the new vertex is at $ (a + c, 4) $ where $ c>0 $. So $ a $ can be any real number, but if we need a possible value, let's say $ a = 2 $ (any real number is possible, so we can choose a value like 2, or 0, or -1, etc.).
# Answer: A possible value of $ a $ is 2 (any real number is acceptable, this is just an example)

### Eighth Question (Value of $ c $)
# Explanation:
## Step1: Combine the Polynomials
We have $ (x^4 + 2x^3 + 5x - 1) + (3x^3 - x^2 + 3) $.
## Step2: Combine Like Terms
For $ x^3 $ terms: $ 2x^3 + 3x^3 = 5x^3 $. So the combined polynomial is $ x^4 + 5x^3 - x^2 + 5x + 2 $. Comparing to $ x^4 + cx^3 - x^2 + 5x + 2 $, we see $ c = 5 $.
# Answer: 5

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QUESTION IMAGE

Question

Explanation:

First Question (Rectangle Width)

Step1: Recall Area Formula

Step2: Perform Polynomial Division

Step1: Simplify the Numerator and Denominator

Step1: Solve the Quadratic Equation

Step2: Calculate \( a + b \)

Answer:

Second Question (Equivalent Rational Expression)