Question

name____________________________ teacher __________ hour ____________ a
9.) factor $x^2 - 5x - 50$

answer:______________________________
10.) factor $7x^2 - 8x - 12$

answer:______________________________
11.) use multiple methods, factor the polynomial completely $36x^2 - 108x + 81$

answer:______________________________

Explanation:

Problem 9: Factor \( x^2 - 5x - 50 \)

Step1: Find two numbers

We need two numbers that multiply to \( -50 \) and add up to \( -5 \). The numbers are \( -10 \) and \( 5 \) because \( -10 \times 5 = -50 \) and \( -10 + 5 = -5 \).

Step2: Factor the quadratic

Using these numbers, we can factor the quadratic as \( (x - 10)(x + 5) \).

Step1: Multiply \( a \) and \( c \)

For the quadratic \( ax^2 + bx + c \), here \( a = 7 \), \( b = -8 \), \( c = -12 \). Multiply \( a \) and \( c \): \( 7 \times (-12) = -84 \).

Step2: Find two numbers

Find two numbers that multiply to \( -84 \) and add up to \( -8 \). The numbers are \( -14 \) and \( 6 \) because \( -14 \times 6 = -84 \) and \( -14 + 6 = -8 \).

Step3: Split the middle term

Rewrite the middle term using these numbers: \( 7x^2 - 14x + 6x - 12 \).

Step4: Factor by grouping

Group the first two and last two terms: \( (7x^2 - 14x) + (6x - 12) \). Factor out the GCF from each group: \( 7x(x - 2) + 6(x - 2) \). Then factor out \( (x - 2) \): \( (7x + 6)(x - 2) \).

Step1: Factor out the GCF

The GCF of \( 36 \), \( -108 \), and \( 81 \) is \( 9 \). Factor out \( 9 \): \( 9(4x^2 - 12x + 9) \).

Step2: Factor the quadratic inside the parentheses

For \( 4x^2 - 12x + 9 \), it is a perfect square trinomial. \( (2x)^2 - 2\times2x\times3 + 3^2 \), which factors as \( (2x - 3)^2 \).

Step3: Combine the factors

Putting it all together, we get \( 9(2x - 3)^2 \).

Answer:

\( (x - 10)(x + 5) \)

Problem 10: Factor \( 7x^2 - 8x - 12 \)