Finite Math Examples

$(3 x + 4) (x + 5) = 3 (x + 2) - 2$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $3 (x + 2)$ from both sides of the equation.

$(3 x + 4) (x + 5) - 3 (x + 2) = - 2$

Step 1.2

Add $2$ to both sides of the equation.

$(3 x + 4) (x + 5) - 3 (x + 2) + 2 = 0$

Step 2

Simplify $(3 x + 4) (x + 5) - 3 (x + 2) + 2$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Expand $(3 x + 4) (x + 5)$ using the FOIL Method.

Tap for more steps...

Step 2.1.1.1

Apply the distributive property.

$3 x (x + 5) + 4 (x + 5) - 3 (x + 2) + 2 = 0$

Step 2.1.1.2

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 5 + 4 (x + 5) - 3 (x + 2) + 2 = 0$

Step 2.1.1.3

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 5 + 4 x + 4 \cdot 5 - 3 (x + 2) + 2 = 0$

Step 2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.1.2.1.1.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot 5 + 4 x + 4 \cdot 5 - 3 (x + 2) + 2 = 0$

Step 2.1.2.1.1.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot 5 + 4 x + 4 \cdot 5 - 3 (x + 2) + 2 = 0$

Step 2.1.2.1.2

Multiply $5$ by $3$ .

$3 x^{2} + 15 x + 4 x + 4 \cdot 5 - 3 (x + 2) + 2 = 0$

Step 2.1.2.1.3

Multiply $4$ by $5$ .

$3 x^{2} + 15 x + 4 x + 20 - 3 (x + 2) + 2 = 0$

Step 2.1.2.2

Add $15 x$ and $4 x$ .

$3 x^{2} + 19 x + 20 - 3 (x + 2) + 2 = 0$

Step 2.1.3

Apply the distributive property.

$3 x^{2} + 19 x + 20 - 3 x - 3 \cdot 2 + 2 = 0$

Step 2.1.4

Multiply $- 3$ by $2$ .

$3 x^{2} + 19 x + 20 - 3 x - 6 + 2 = 0$

Step 2.2

Subtract $3 x$ from $19 x$ .

$3 x^{2} + 16 x + 20 - 6 + 2 = 0$

Step 2.3

Subtract $6$ from $20$ .

$3 x^{2} + 16 x + 14 + 2 = 0$

Step 2.4

Add $14$ and $2$ .

$3 x^{2} + 16 x + 16 = 0$

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

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 $a \cdot c = 3 \cdot 16 = 48$ and whose sum is $b = 16$ .

Tap for more steps...

Step 3.1.1

Factor $16$ out of $16 x$ .

$3 x^{2} + 16 (x) + 16 = 0$

Step 3.1.2

Rewrite $16$ as $4$ plus $12$

$3 x^{2} + (4 + 12) x + 16 = 0$

Step 3.1.3

Apply the distributive property.

$3 x^{2} + 4 x + 12 x + 16 = 0$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$(3 x^{2} + 4 x) + 12 x + 16 = 0$

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

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

Step 4

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

$3 x + 4 = 0$

$x + 4 = 0$

Step 5

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

Tap for more steps...

Step 5.1

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

$3 x + 4 = 0$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 5.2.2

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

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.2.3.1

Move the negative in front of the fraction.

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

Step 6

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

Tap for more steps...

Step 6.1

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

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

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

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