Solve for x square root of (x+5)^2=x+5

$\sqrt{{(x + 5)}^{2}} = x + 5$

Step 1

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

${\sqrt{{(x + 5)}^{2}}}^{2} = {(x + 5)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(x + 5)}^{2}}$ as ${(x + 5)}^{\frac{2}{2}}$ .

${({(x + 5)}^{\frac{2}{2}})}^{2} = {(x + 5)}^{2}$

Divide $2$ by $2$ .

${({(x + 5)}^{1})}^{2} = {(x + 5)}^{2}$

Simplify the left side.

Multiply the exponents in ${({(x + 5)}^{1})}^{2}$ .

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

${(x + 5)}^{1 \cdot 2} = {(x + 5)}^{2}$

Multiply $2$ by $1$ .

${(x + 5)}^{2} = {(x + 5)}^{2}$

Simplify the right side.

Simplify ${(x + 5)}^{2}$ .

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

${(x + 5)}^{2} = (x + 5) (x + 5)$

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

Apply the distributive property.

${(x + 5)}^{2} = x (x + 5) + 5 (x + 5)$

Apply the distributive property.

${(x + 5)}^{2} = x \cdot x + x \cdot 5 + 5 (x + 5)$

Apply the distributive property.

${(x + 5)}^{2} = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

${(x + 5)}^{2} = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5$

Move $5$ to the left of $x$ .

${(x + 5)}^{2} = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5$

Multiply $5$ by $5$ .

${(x + 5)}^{2} = x^{2} + 5 x + 5 x + 25$

Add $5 x$ and $5 x$ .

${(x + 5)}^{2} = x^{2} + 10 x + 25$

Step 3

Solve for $x$ .

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

$x^{2} + 10 x + 25 = {(x + 5)}^{2}$

Simplify ${(x + 5)}^{2}$ .

Rewrite.

$x^{2} + 10 x + 25 = 0 + 0 + {(x + 5)}^{2}$

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$x^{2} + 10 x + 25 = (x + 5) (x + 5)$

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

Apply the distributive property.

$x^{2} + 10 x + 25 = x (x + 5) + 5 (x + 5)$

Apply the distributive property.

$x^{2} + 10 x + 25 = x \cdot x + x \cdot 5 + 5 (x + 5)$

Apply the distributive property.

$x^{2} + 10 x + 25 = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 10 x + 25 = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5$

Move $5$ to the left of $x$ .

$x^{2} + 10 x + 25 = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$x^{2} + 10 x + 25 = x^{2} + 5 x + 5 x + 25$

Add $5 x$ and $5 x$ .

$x^{2} + 10 x + 25 = x^{2} + 10 x + 25$

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

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 10 x + 25 - x^{2} = 10 x + 25$

Subtract $10 x$ from both sides of the equation.

$x^{2} + 10 x + 25 - x^{2} - 10 x = 25$

Combine the opposite terms in $x^{2} + 10 x + 25 - x^{2} - 10 x$ .

Subtract $x^{2}$ from $x^{2}$ .

$10 x + 25 + 0 - 10 x = 25$

Add $10 x + 25$ and $0$ .

$10 x + 25 - 10 x = 25$

Subtract $10 x$ from $10 x$ .

$0 + 25 = 25$

Add $0$ and $25$ .

$25 = 25$

Since $25 = 25$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve for x square root of (x+5)^2=x+5

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