Solve for x 3(x^2+x)=2(x+2)

$3 (x^{2} + x) = 2 (x + 2)$

Step 1

Simplify $3 (x^{2} + x)$ .

Start simplifying.

$0 + 0 + 3 (x^{2} + x) = 2 (x + 2)$

Simplify by adding zeros.

$3 (x^{2} + x) = 2 (x + 2)$

Apply the distributive property.

$3 x^{2} + 3 x = 2 (x + 2)$

Step 2

Simplify $2 (x + 2)$ .

Apply the distributive property.

$3 x^{2} + 3 x = 2 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$3 x^{2} + 3 x = 2 x + 4$

Step 3

Move all terms containing $x$ to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$3 x^{2} + 3 x - 2 x = 4$

Subtract $2 x$ from $3 x$ .

$3 x^{2} + x = 4$

Step 4

Subtract $4$ from both sides of the equation.

$3 x^{2} + x - 4 = 0$

Step 5

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

Multiply by $1$ .

$3 x^{2} + 1 x - 4 = 0$

Rewrite $1$ as $- 3$ plus $4$

$3 x^{2} + (- 3 + 4) x - 4 = 0$

Apply the distributive property.

$3 x^{2} - 3 x + 4 x - 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} - 3 x) + 4 x - 4 = 0$

Factor out the greatest common factor (GCF) from each group.

$3 x (x - 1) + 4 (x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (3 x + 4) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$3 x + 4 = 0$

Step 7

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 8

Set $3 x + 4$ equal to $0$ and solve for $x$ .

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Solve $3 x + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Divide each term in $3 x = - 4$ by $3$ and simplify.

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 9

The final solution is all the values that make $(x - 1) (3 x + 4) = 0$ true.

$x = 1, - \frac{4}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 1, - \frac{4}{3}$

Decimal Form:

$x = 1, - 1.\overline{3}$

Mixed Number Form:

$x = 1, - 1 \frac{1}{3}$

Solve for x 3(x^2+x)=2(x+2)

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