Question

1. factor the following:

a) $9x^2 - 4$
b) $2x^2 - 98$
c) $x^2 - 13x - 48$
d) $x^2 - 9x + 14$

1. are the triangles similar? if so, by what

Explanation:

Part a: Factor \(9x^2 - 4\)

Step 1: Recognize as difference of squares (\(a^2 - b^2=(a - b)(a + b)\))
Here, \(a = 3x\), \(b = 2\) (since \((3x)^2=9x^2\) and \(2^2 = 4\))
Step 2: Apply formula
\(9x^2-4=(3x - 2)(3x + 2)\)

Part b: Factor \(2x^2-98\)

Step 1: Factor out GCF (2)
\(2x^2-98=2(x^2 - 49)\)
Step 2: Recognize \(x^2 - 49\) as difference of squares (\(x^2-7^2\))
Step 3: Apply formula \(a^2 - b^2=(a - b)(a + b)\)
\(2(x^2 - 49)=2(x - 7)(x + 7)\)

Part c: Factor \(x^2-13x - 48\)

Step 1: Find two numbers that multiply to \(- 48\) and add to \(-13\)
Numbers: \(-16\) and \(3\) (since \(-16\times3=-48\) and \(-16 + 3=-13\))
Step 2: Split middle term and factor by grouping
\(x^2-16x+3x - 48=x(x - 16)+3(x - 16)=(x - 16)(x + 3)\)

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

Step 1: Find two numbers that multiply to \(14\) and add to \(-9\)
Numbers: \(-7\) and \(-2\) (since \((-7)\times(-2)=14\) and \(-7+(-2)=-9\))
Step 2: Split middle term and factor by grouping
\(x^2-7x-2x + 14=x(x - 7)-2(x - 7)=(x - 7)(x - 2)\)

Answer:

a) \(\boldsymbol{(3x - 2)(3x + 2)}\)
b) \(\boldsymbol{2(x - 7)(x + 7)}\)
c) \(\boldsymbol{(x - 16)(x + 3)}\)
d) \(\boldsymbol{(x - 7)(x - 2)}\)