Question

day 4
factor completely: $x^2 - 9$
simplify: $(x^2 + 3x)(x - 2)$
day 5
find the slope of the line passing through $(1,2)$ and $(3,4)$
solve absolute value equation: $|x - 5| = 7$

Explanation:

Day 4 - Factor completely: \( x^2 - 9 \)

Step 1: Identify the formula

The expression \( x^2 - 9 \) is a difference of squares, which follows the formula \( a^2 - b^2=(a + b)(a - b) \). Here, \( a = x \) and \( b = 3 \) (since \( 9=3^2 \)).

Step 2: Apply the formula

Substitute \( a = x \) and \( b = 3 \) into the difference of squares formula: \( x^2-9=x^2 - 3^2=(x + 3)(x - 3) \).

Step 1: Distribute the terms

Use the distributive property (FOIL method for binomials, extended here). Multiply \( x^2 \) by \( (x - 2) \) and \( 3x \) by \( (x - 2) \):
\( (x^2+3x)(x - 2)=x^2(x - 2)+3x(x - 2) \)

Step 2: Expand each product

Expand \( x^2(x - 2) \) to get \( x^3-2x^2 \), and expand \( 3x(x - 2) \) to get \( 3x^2-6x \).

Step 3: Combine like terms

Add the expanded terms together: \( x^3-2x^2+3x^2-6x=x^3+( - 2x^2+3x^2)-6x=x^3+x^2-6x \)

Step 1: Recall the slope formula

The slope \( m \) of a line passing through two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Substitute the values

Here, \( x_1 = 1,y_1 = 2,x_2 = 3,y_2 = 4 \). Substitute into the formula: \( m=\frac{4 - 2}{3 - 1}=\frac{2}{2}=1 \)

Answer:

\( (x + 3)(x - 3) \)

Day 4 - Simplify: \( (x^2 + 3x)(x - 2) \)