Question

unit 7: factoring practice packet
complete the following questions and show all of your work in order to receive full credit.

1. factor using the big x and t chart..

$x^2 + 9x - 10$

1. factor using the big x and t chart.

$6x^2 + x - 12$

1. factor using the big x and t chart.

$4x^2 - 16$

1. factor using the big x and t chart.

$- 3x^2 + 19x - 20$

Explanation:

Problem 1: Factor \( x^2 + 9x - 10 \)

Step 1: Identify \( a \), \( b \), \( c \)

For \( ax^2 + bx + c \), here \( a = 1 \), \( b = 9 \), \( c = -10 \). We need two numbers that multiply to \( a \times c = 1 \times (-10) = -10 \) and add to \( b = 9 \).
The numbers are \( 10 \) and \( -1 \) (since \( 10 \times (-1) = -10 \) and \( 10 + (-1) = 9 \)).

Step 2: Rewrite the middle term

Split \( 9x \) into \( 10x - x \):
\( x^2 + 10x - x - 10 \)

Step 3: Group and factor

Group the first two and last two terms:
\( (x^2 + 10x) + (-x - 10) \)
Factor out the GCF from each group:
\( x(x + 10) - 1(x + 10) \)
Factor out \( (x + 10) \):
\( (x + 10)(x - 1) \)

Step 1: Identify \( a \), \( b \), \( c \)

Here \( a = 6 \), \( b = 1 \), \( c = -12 \). Calculate \( a \times c = 6 \times (-12) = -72 \). Find two numbers that multiply to \( -72 \) and add to \( 1 \). The numbers are \( 9 \) and \( -8 \) (since \( 9 \times (-8) = -72 \) and \( 9 + (-8) = 1 \)).

Step 2: Rewrite the middle term

Split \( x \) into \( 9x - 8x \):
\( 6x^2 + 9x - 8x - 12 \)

Step 3: Group and factor

Group the first two and last two terms:
\( (6x^2 + 9x) + (-8x - 12) \)
Factor out GCF from each group:
\( 3x(2x + 3) - 4(2x + 3) \)
Factor out \( (2x + 3) \):
\( (3x - 4)(2x + 3) \)

Step 1: Factor out the GCF

The GCF of \( 4x^2 \) and \( -16 \) is \( 4 \):
\( 4(x^2 - 4) \)

Step 2: Factor the difference of squares

\( x^2 - 4 \) is a difference of squares (\( x^2 - 2^2 \)), which factors as \( (x - 2)(x + 2) \).

Answer:

\( (x + 10)(x - 1) \)

Problem 2: Factor \( 6x^2 + x - 12 \)