Algebra Examples

$2 x (x + 3) + 2 (x + 3) = 0$

Step 1

Simplify $2 x (x + 3) + 2 (x + 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$2 x \cdot x + 2 x \cdot 3 + 2 (x + 3) = 0$

Step 1.1.2

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

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Step 1.1.2.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot 3 + 2 (x + 3) = 0$

Step 1.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3

Multiply $3$ by $2$ .

$2 x^{2} + 6 x + 2 (x + 3) = 0$

Step 1.1.4

Apply the distributive property.

$2 x^{2} + 6 x + 2 x + 2 \cdot 3 = 0$

Step 1.1.5

Multiply $2$ by $3$ .

$2 x^{2} + 6 x + 2 x + 6 = 0$

Step 1.2

Add $6 x$ and $2 x$ .

$2 x^{2} + 8 x + 6 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $2$ out of $2 x^{2} + 8 x + 6$ .

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Step 2.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 8 x + 6 = 0$

Step 2.1.2

Factor $2$ out of $8 x$ .

$2 (x^{2}) + 2 (4 x) + 6 = 0$

Step 2.1.3

Factor $2$ out of $6$ .

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

Step 2.1.4

Factor $2$ out of $2 x^{2} + 2 (4 x)$ .

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

Step 2.1.5

Factor $2$ out of $2 (x^{2} + 4 x) + 2 \cdot 3$ .

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

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.2.1.2

Write the factored form using these integers.

$2 ((x + 1) (x + 3)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$2 (x + 1) (x + 3) = 0$

Step 3

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

$x + 1 = 0$

$x + 3 = 0$

Step 4

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

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

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

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Step 5.1

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

$x + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

The final solution is all the values that make $2 (x + 1) (x + 3) = 0$ true.

$x = - 1, - 3$