for problems 1 - 10, rewrite the expression in factored form.
1. $14x^2 + 2x$
2. $x^2 - 25$
3. $16x^2 - 36$
4. $3x^2 - 27$
5. $x^2 + 10x + 25$
6. $x^2 - 3x + 2$
7. $x^2 + x - 90$

Question

Accepted Answer

### Problem 1: $14x^2 + 2x$
# Explanation:
## Step1: Find the GCF of $14x^2$ and $2x$
The GCF of $14$ and $2$ is $2$, and the GCF of $x^2$ and $x$ is $x$. So the GCF is $2x$.
## Step2: Factor out the GCF
$14x^2 + 2x = 2x(7x + 1)$
# Answer:
$2x(7x + 1)$

### Problem 2: $x^2 - 25$
# Explanation:
## Step1: Recognize the difference of squares
The expression is in the form $a^2 - b^2$ where $a = x$ and $b = 5$ (since $25 = 5^2$).
## Step2: Apply the difference of squares formula $a^2 - b^2=(a + b)(a - b)$
$x^2 - 25=(x + 5)(x - 5)$
# Answer:
$(x + 5)(x - 5)$

### Problem 3: $16x^2 - 36$
# Explanation:
## Step1: Find the GCF of $16x^2$ and $36$
The GCF of $16$ and $36$ is $4$.
## Step2: Factor out the GCF
$16x^2 - 36 = 4(4x^2 - 9)$
## Step3: Recognize the difference of squares in $4x^2 - 9$
Here, $a = 2x$ and $b = 3$ (since $4x^2=(2x)^2$ and $9 = 3^2$).
## Step4: Apply the difference of squares formula
$4x^2 - 9=(2x + 3)(2x - 3)$, so $16x^2 - 36 = 4(2x + 3)(2x - 3)$
# Answer:
$4(2x + 3)(2x - 3)$

### Problem 4: $3x^2 - 27$
# Explanation:
## Step1: Find the GCF of $3x^2$ and $27$
The GCF is $3$.
## Step2: Factor out the GCF
$3x^2 - 27 = 3(x^2 - 9)$
## Step3: Recognize the difference of squares in $x^2 - 9$
Here, $a = x$ and $b = 3$ (since $9 = 3^2$).
## Step4: Apply the difference of squares formula
$x^2 - 9=(x + 3)(x - 3)$, so $3x^2 - 27 = 3(x + 3)(x - 3)$
# Answer:
$3(x + 3)(x - 3)$

### Problem 5: $x^2 + 10x + 25$
# Explanation:
## Step1: Recognize the perfect square trinomial
The expression is in the form $a^2 + 2ab + b^2$ where $a = x$ and $b = 5$ (since $2ab=2\times x\times5 = 10x$ and $b^2 = 25$).
## Step2: Apply the perfect square trinomial formula $a^2 + 2ab + b^2=(a + b)^2$
$x^2 + 10x + 25=(x + 5)^2$
# Answer:
$(x + 5)^2$

### Problem 6: $x^2 - 3x + 2$
# Explanation:
## Step1: Find two numbers that multiply to $2$ and add to $-3$
The numbers are $-1$ and $-2$ (since $(-1)\times(-2)=2$ and $(-1)+(-2)=-3$).
## Step2: Factor the trinomial
$x^2 - 3x + 2=(x - 1)(x - 2)$
# Answer:
$(x - 1)(x - 2)$

### Problem 7: $x^2 + x - 90$
# Explanation:
## Step1: Find two numbers that multiply to $-90$ and add to $1$
The numbers are $10$ and $-9$ (since $10\times(-9)=-90$ and $10+(-9)=1$).
## Step2: Factor the trinomial
$x^2 + x - 90=(x + 10)(x - 9)$
# Answer:
$(x + 10)(x - 9)$

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QUESTION IMAGE

Question

Explanation:

Problem 1: \(14x^2 + 2x\)

Step1: Find the GCF of \(14x^2\) and \(2x\)

Step2: Factor out the GCF

Step1: Recognize the difference of squares

Step2: Apply the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\)

Step1: Find the GCF of \(16x^2\) and \(36\)

Step2: Factor out the GCF

Step3: Recognize the difference of squares in \(4x^2 - 9\)

Step4: Apply the difference of squares formula

Answer:

Problem 2: \(x^2 - 25\)