Question

36 factor the following:
a) $9x^2 - 4$ (with handwritten $(3x - 2)(3x + 2)$)
b) $2x^2 - 98$
c) $x^2 - 13x - 48$ (with some handwritten notes)
d) $x^2 - 9x + 14$
38 are the triangles similar? if so by what

Explanation:

Part a) Factoring \(9x^2 - 4\)

Step 1: Identify the form

The expression \(9x^2 - 4\) is a difference of squares, since \(9x^2=(3x)^2\) and \(4 = 2^2\). The formula for factoring a difference of squares is \(a^2 - b^2=(a - b)(a + b)\).

Step 2: Apply the formula

Here, \(a = 3x\) and \(b = 2\). So, \(9x^2-4=(3x - 2)(3x + 2)\).

Part b) Factoring \(2x^2-98\)

Step 1: Factor out the GCF

First, we can factor out the greatest common factor (GCF) of \(2x^2\) and \(98\), which is \(2\). So, \(2x^2-98=2(x^2 - 49)\).

Step 2: Factor the difference of squares

The expression \(x^2 - 49\) is also a difference of squares, with \(a=x\) and \(b = 7\) (since \(49=7^2\)). Using the difference of squares formula \(a^2 - b^2=(a - b)(a + b)\), we get \(x^2-49=(x - 7)(x + 7)\). So, \(2x^2-98=2(x - 7)(x + 7)\).

Part c) Factoring \(x^2-13x - 48\)

Step 1: Find two numbers

We need two numbers that multiply to \(- 48\) and add up to \(-13\). Let's list the factor pairs of \(-48\):

• \(1\times(-48)=-48\), and \(1+(-48)=-47\)
• \(2\times(-24)=-48\), and \(2+(-24)=-22\)
• \(3\times(-16)=-48\), and \(3+(-16)=-13\)

Step 2: Write the factored form

Using the two numbers \(3\) and \(-16\), we can factor the quadratic as \((x + 3)(x-16)\) (because \((x + 3)(x - 16)=x^2-16x+3x - 48=x^2-13x - 48\)).

Part d) Factoring \(x^2-9x + 14\)

Answer:

Step 1: Find two numbers

We need two numbers that multiply to \(14\) and add up to \(-9\). The factor pairs of \(14\) are:

• \(1\times14 = 14\), and \(1 + 14=15\)
• \((-1)\times(-14)=14\), and \(-1+(-14)=-15\)
• \(2\times7 = 14\), and \(2 + 7 = 9\)
• \((-2)\times(-7)=14\), and \(-2+(-7)=-9\)

Step 2: Write the factored form

Using the two numbers \(-2\) and \(-7\), we can factor the quadratic as \((x - 2)(x - 7)\) (because \((x - 2)(x - 7)=x^2-7x-2x + 14=x^2-9x + 14\)).

Final Answers:

a) \(\boldsymbol{(3x - 2)(3x + 2)}\)

b) \(\boldsymbol{2(x - 7)(x + 7)}\)

c) \(\boldsymbol{(x + 3)(x - 16)}\)

d) \(\boldsymbol{(x - 2)(x - 7)}\)