Factor completely. Then, solve. (Hint: Find GCF) $$ y=5 x^{2}+35 x+50 $$

Question

UpStudy AI · Accepted Answer

To factor the quadratic expression $$ y = 5x^{2} + 35x + 50 $$ completely and solve for $$ x $$, follow these steps:

### Step 1: Factor Out the Greatest Common Factor (GCF)

First, identify the GCF of the coefficients $$ 5 $$, $$ 35 $$, and $$ 50 $$, which is **5**.

$$
y = 5x^{2} + 35x + 50 = 5(x^{2} + 7x + 10)
$$

### Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression inside the parentheses:

$$
x^{2} + 7x + 10
$$

Look for two numbers that multiply to **10** (the constant term) and add up to **7** (the coefficient of $$ x $$). These numbers are **2** and **5**.

$$
x^{2} + 7x + 10 = (x + 2)(x + 5)
$$

### Step 3: Complete Factoring

Combine the factored terms with the GCF:

$$
y = 5(x + 2)(x + 5)
$$

### Step 4: Solve for $$ x $$ When $$ y = 0 $$

Set the equation equal to zero and solve for $$ x $$:

$$
5(x + 2)(x + 5) = 0
$$

Divide both sides by 5:

$$
(x + 2)(x + 5) = 0
$$

Set each factor equal to zero:

$$
x + 2 = 0 \quad \text{or} \quad x + 5 = 0
$$

$$
x = -2 \quad \text{or} \quad x = -5
$$

### Final Answer

$$
\boxed{ y = 5(x + 2)(x + 5) \quad \text{with solutions}\ x = -2 \ \text{and}\ x = -5 }
$$
The quadratic expression $$ y = 5x^{2} + 35x + 50 $$ factors to $$ y = 5(x + 2)(x + 5) $$. The solutions for $$ x $$ are $$ x = -2 $$ and $$ x = -5 $$.

Factor completely. Then, solve. (Hint: Find GCF)

Upstudy AI Solution

Answer

Solution

Step 1: Factor Out the Greatest Common Factor (GCF)

Step 2: Factor the Quadratic Expression

Step 3: Complete Factoring

Step 4: Solve for When

Final Answer

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