Question

question 2 (2 points)
solve the equation $3(x + 7) - 2 = 10x - 16$ for $x$. you must show your work for full credit

question 3 (4 points)
completely factor each expression:

a. $2x^3y^2 + 8x^5y^3$

b. $9x^2 - 16y^4$

c. $3x^2 - 15x + 18$

d. $3x^3 - 12x$

Explanation:

Question 2

Step 1: Expand the left - hand side

We use the distributive property \(a(b + c)=ab+ac\) to expand \(5(x + 7)-2\). First, expand \(5(x + 7)\): \(5(x + 7)=5x+35\). Then subtract 2: \(5x + 35-2=5x+33\). So the equation becomes \(5x + 33=10x-16\).

Step 2: Move the \(x\) terms to one side

Subtract \(5x\) from both sides of the equation: \(5x+33 - 5x=10x-16 - 5x\). This simplifies to \(33 = 5x-16\).

Step 3: Solve for \(x\)

Add 16 to both sides: \(33+16=5x-16 + 16\). So \(49 = 5x\). Then divide both sides by 5: \(x=\frac{49}{5}=9.8\).

Step 1: Find the GCF

For the expression \(2x^{3}y^{2}+8x^{5}y^{3}\), we find the greatest common factor (GCF) of the coefficients and the variables. The GCF of 2 and 8 is 2. For the \(x\) - terms, the GCF of \(x^{3}\) and \(x^{5}\) is \(x^{3}\). For the \(y\) - terms, the GCF of \(y^{2}\) and \(y^{3}\) is \(y^{2}\). So the GCF is \(2x^{3}y^{2}\).

Step 2: Factor out the GCF

Factor out \(2x^{3}y^{2}\) from each term: \(2x^{3}y^{2}(1 + 4x^{2}y)\).

Step 1: Recognize the difference of squares

The expression \(9x^{2}-16y^{4}\) can be written as \((3x)^{2}-(4y^{2})^{2}\). The difference of squares formula is \(a^{2}-b^{2}=(a + b)(a - b)\).

Step 2: Apply the difference of squares formula

Let \(a = 3x\) and \(b = 4y^{2}\). Then, using the difference of squares formula, we get \((3x + 4y^{2})(3x-4y^{2})\).

Answer:

\(x = \frac{49}{5}\) (or \(x = 9.8\))

Question 3

Part a