Questions: Solve the equation by factoring. 3y^2 + 16y + 5 = 0 y= (smaller value) y= (larger value)

Transcript text: Solve the equation by factoring. \[ 3 y^{2}+16 y+5=0 \] $y=$ $\square$ (smaller value) $y=$ $\square$ (larger value)

Solution

Solution Steps

To solve the quadratic equation $3y^2 + 16y + 5 = 0$ by factoring, we need to find two numbers that multiply to $3 \times 5 = 15$ and add up to $16$. Once we find these numbers, we can rewrite the middle term and factor by grouping.

Step 1: Identify the quadratic equation

We start with the quadratic equation: \[ 3y^2 + 16y + 5 = 0 \]

Step 2: Factor the quadratic equation

To factor the quadratic equation, we need to find two numbers that multiply to $3 \times 5 = 15$ and add up to $16$. These numbers are $15$ and $1$.

Step 3: Rewrite the middle term and factor by grouping

Rewrite the equation as: \[ 3y^2 + 15y + y + 5 = 0 \] Group the terms: \[ (3y^2 + 15y) + (y + 5) = 0 \] Factor out the common terms: \[ 3y(y + 5) + 1(y + 5) = 0 \] Factor out the common binomial: \[ (y + 5)(3y + 1) = 0 \]

Step 4: Solve for $y$

Set each factor equal to zero and solve for $y$: \[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \] \[ 3y + 1 = 0 \quad \Rightarrow \quad y = -\frac{1}{3} \]