directions: factor each expression. be sure to che
7. ( x^2 + 10x + 24 )
9. ( 6p^2 - 24p + 24 ) (with handwritten ( 6(p^2 - 4p + 4) ))
11. ( 2x^2 - 15x - 8 )
13. ( 16w^2 + 8w + 1 )
15. ( k^2 - 25 )
17. ( 64x^2y - y )

Question

Accepted Answer

### Problem 7: $ x^2 + 10x + 24 $
# Explanation:
## Step1: Find two numbers that multiply to 24 and add to 10.
The numbers are 4 and 6, since $ 4 \times 6 = 24 $ and $ 4 + 6 = 10 $.
## Step2: Factor the quadratic.
Using the numbers from Step 1, we can write the quadratic as $ (x + 4)(x + 6) $.
# Answer:
$ (x + 4)(x + 6) $

### Problem 9: $ 6p^2 - 24p + 24 $
# Explanation:
## Step1: Factor out the greatest common factor (GCF).
The GCF of 6, -24, and 24 is 6. So, factor out 6: $ 6(p^2 - 4p + 4) $.
## Step2: Factor the quadratic inside the parentheses.
The quadratic $ p^2 - 4p + 4 $ is a perfect square trinomial, which factors as $ (p - 2)^2 $.
## Step3: Combine the factors.
Putting it all together, we get $ 6(p - 2)^2 $.
# Answer:
$ 6(p - 2)^2 $

### Problem 11: $ 2x^2 - 15x - 8 $
# Explanation:
## Step1: Multiply the coefficient of $ x^2 $ and the constant term.
$ 2 \times (-8) = -16 $.
## Step2: Find two numbers that multiply to -16 and add to -15.
The numbers are -16 and 1, since $ -16 \times 1 = -16 $ and $ -16 + 1 = -15 $.
## Step3: Rewrite the middle term using these numbers.
$ 2x^2 - 16x + x - 8 $.
## Step4: Factor by grouping.
Group the first two terms and the last two terms: $ (2x^2 - 16x) + (x - 8) $.
Factor out the GCF from each group: $ 2x(x - 8) + 1(x - 8) $.
## Step5: Factor out the common binomial factor.
$ (2x + 1)(x - 8) $.
# Answer:
$ (2x + 1)(x - 8) $

### Problem 13: $ 16w^2 + 8w + 1 $
# Explanation:
## Step1: Recognize the perfect square trinomial.
The first term $ 16w^2 = (4w)^2 $, the last term $ 1 = 1^2 $, and the middle term $ 8w = 2 \times 4w \times 1 $.
## Step2: Factor the perfect square trinomial.
Using the formula $ a^2 + 2ab + b^2 = (a + b)^2 $, we get $ (4w + 1)^2 $.
# Answer:
$ (4w + 1)^2 $

### Problem 15: $ k^2 - 25 $
# Explanation:
## Step1: Recognize the difference of squares.
The expression is in the form $ a^2 - b^2 $, where $ a = k $ and $ b = 5 $ (since $ 25 = 5^2 $).
## Step2: Factor the difference of squares.
Using the formula $ a^2 - b^2 = (a + b)(a - b) $, we get $ (k + 5)(k - 5) $.
# Answer:
$ (k + 5)(k - 5) $

### Problem 17: $ 64x^2y - y $
# Explanation:
## Step1: Factor out the GCF.
The GCF of $ 64x^2y $ and $ -y $ is $ y $. So, factor out $ y $: $ y(64x^2 - 1) $.
## Step2: Recognize the difference of squares.
The expression $ 64x^2 - 1 $ is in the form $ a^2 - b^2 $, where $ a = 8x $ (since $ 64x^2 = (8x)^2 $) and $ b = 1 $ (since $ 1 = 1^2 $).
## Step3: Factor the difference of squares.
Using the formula $ a^2 - b^2 = (a + b)(a - b) $, we get $ y(8x + 1)(8x - 1) $.
# Answer:
$ y(8x + 1)(8x - 1) $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7: \( x^2 + 10x + 24 \)

Step1: Find two numbers that multiply to 24 and add to 10.

Step2: Factor the quadratic.

Step1: Factor out the greatest common factor (GCF).

Step2: Factor the quadratic inside the parentheses.

Step3: Combine the factors.

Step1: Multiply the coefficient of \( x^2 \) and the constant term.

Step2: Find two numbers that multiply to -16 and add to -15.

Step3: Rewrite the middle term using these numbers.

Step4: Factor by grouping.

Step5: Factor out the common binomial factor.

Answer:

Problem 9: \( 6p^2 - 24p + 24 \)