Algebra Examples

$\sqrt{4 + x} = - 2 - x$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 + x}}^{2} = {(- 2 - x)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + x}$ as ${(4 + x)}^{\frac{1}{2}}$ .

${({(4 + x)}^{\frac{1}{2}})}^{2} = {(- 2 - x)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 + x)}^{\frac{1}{2} \cdot 2} = {(- 2 - x)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(4 + x)}^{\frac{1}{2} \cdot 2} = {(- 2 - x)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 + x)}^{1} = {(- 2 - x)}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(- 2 - x)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(- 2 - x)}^{2}$ as $(- 2 - x) (- 2 - x)$ .

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

Step 2.3.1.2

Expand $(- 2 - x) (- 2 - x)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

$4 + x = - 2 \cdot - 2 - 2 (- x) - x (- 2 - x)$

Step 2.3.1.2.3

Apply the distributive property.

$4 + x = - 2 \cdot - 2 - 2 (- x) - x \cdot - 2 - x (- x)$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $- 2$ by $- 2$ .

$4 + x = 4 - 2 (- x) - x \cdot - 2 - x (- x)$

Step 2.3.1.3.1.2

Multiply $- 1$ by $- 2$ .

$4 + x = 4 + 2 x - x \cdot - 2 - x (- x)$

Step 2.3.1.3.1.3

Multiply $- 2$ by $- 1$ .

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

Step 2.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 + x = 4 + 2 x + 2 x - 1 \cdot - 1 x \cdot x$

Step 2.3.1.3.1.5

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

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Step 2.3.1.3.1.5.1

Move $x$ .

$4 + x = 4 + 2 x + 2 x - 1 \cdot - 1 (x \cdot x)$

Step 2.3.1.3.1.5.2

Multiply $x$ by $x$ .

$4 + x = 4 + 2 x + 2 x - 1 \cdot - 1 x^{2}$

Step 2.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 + x = 4 + 2 x + 2 x + 1 x^{2}$

Step 2.3.1.3.1.7

Multiply $x^{2}$ by $1$ .

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

Step 2.3.1.3.2

Add $2 x$ and $2 x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3.2

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

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Step 3.2.1

Subtract $x$ from both sides of the equation.

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

Step 3.2.2

Subtract $x$ from $4 x$ .

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

Step 3.3

Subtract $4$ from both sides of the equation.

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

Step 3.4

Combine the opposite terms in $4 + x^{2} + 3 x - 4$ .

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Step 3.4.1

Subtract $4$ from $4$ .

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

Step 3.4.2

Add $x^{2} + 3 x$ and $0$ .

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

Step 3.5

Factor $x$ out of $x^{2} + 3 x$ .

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Step 3.5.1

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

$x \cdot x + 3 x = 0$

Step 3.5.2

Factor $x$ out of $3 x$ .

$x \cdot x + x \cdot 3 = 0$

Step 3.5.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

$x (x + 3) = 0$

Step 3.6

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

$x = 0$

$x + 3 = 0$

Step 3.7

Set $x$ equal to $0$ .

$x = 0$

Step 3.8

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

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Step 3.8.1

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

$x + 3 = 0$

Step 3.8.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.9

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

$x = 0, - 3$

Step 4

Exclude the solutions that do not make $\sqrt{4 + x} = - 2 - x$ true.

$x = - 3$