Question

Solve equation by factoring
3x^2+7x+2=0

Answer 1

To solve the equation \( 3x^2 + 7x + 2 = 0 \) by factoring, we first look for two numbers that multiply to \( a \cdot c \) (where \( a \) is the coefficient of \( x^2 \) and \( c \) is the constant term) and add up to \( b \) (the coefficient of \( x \)).

In our equation:
- \( a = 3 \)
- \( b = 7 \)
- \( c = 2 \)

First, we calculate \( a \cdot c \):
\[
3 \cdot 2 = 6
\]

Next, we look for two numbers that multiply to \( 6 \) and add to \( 7 \). Those numbers are \( 6 \) and \( 1 \).

Now, we can rewrite the middle term \( 7x \) using \( 6x \) and \( 1x \):
\[
3x^2 + 6x + 1x + 2 = 0
\]

Next, we group the terms:
\[
(3x^2 + 6x) + (1x + 2) = 0
\]

Now, we factor by grouping:
\[
3x(x + 2) + 1(x + 2) = 0
\]

We can factor out the common term \( (x + 2) \):
\[
(3x + 1)(x + 2) = 0
\]

Now, we set each factor to zero:
1. \( 3x + 1 = 0 \)
2. \( x + 2 = 0 \)

Solving the first equation:
\[
3x + 1 = 0 \implies 3x = -1 \implies x = -\frac{1}{3}
\]

Solving the second equation:
\[
x + 2 = 0 \implies x = -2
\]

Thus, the solutions to the equation \( 3x^2 + 7x + 2 = 0 \) are:
\[
x = -\frac{1}{3} \quad \text{and} \quad x = -2
\]