Solve for a square root of a-4+1 = square root of a+5

$\sqrt{a - 4} + 1 = \sqrt{a + 5}$

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{a - 4} = \sqrt{a + 5} - 1$

Step 2

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

${\sqrt{a - 4}}^{2} = {(\sqrt{a + 5} - 1)}^{2}$

Step 3

Simplify each side of the equation.

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

${({(a - 4)}^{\frac{1}{2}})}^{2} = {(\sqrt{a + 5} - 1)}^{2}$

Simplify the left side.

Simplify ${({(a - 4)}^{\frac{1}{2}})}^{2}$ .

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

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

${(a - 4)}^{\frac{1}{2} \cdot 2} = {(\sqrt{a + 5} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(a - 4)}^{\frac{1}{2} \cdot 2} = {(\sqrt{a + 5} - 1)}^{2}$

Rewrite the expression.

${(a - 4)}^{1} = {(\sqrt{a + 5} - 1)}^{2}$

Simplify.

$a - 4 = {(\sqrt{a + 5} - 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{a + 5} - 1)}^{2}$ .

Rewrite ${(\sqrt{a + 5} - 1)}^{2}$ as $(\sqrt{a + 5} - 1) (\sqrt{a + 5} - 1)$ .

$a - 4 = (\sqrt{a + 5} - 1) (\sqrt{a + 5} - 1)$

Expand $(\sqrt{a + 5} - 1) (\sqrt{a + 5} - 1)$ using the FOIL Method.

Apply the distributive property.

$a - 4 = \sqrt{a + 5} (\sqrt{a + 5} - 1) - 1 (\sqrt{a + 5} - 1)$

Apply the distributive property.

$a - 4 = \sqrt{a + 5} \sqrt{a + 5} + \sqrt{a + 5} \cdot - 1 - 1 (\sqrt{a + 5} - 1)$

Apply the distributive property.

$a - 4 = \sqrt{a + 5} \sqrt{a + 5} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{a + 5} \sqrt{a + 5}$ .

Raise $\sqrt{a + 5}$ to the power of $1$ .

$a - 4 = {\sqrt{a + 5}}^{1} \sqrt{a + 5} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Raise $\sqrt{a + 5}$ to the power of $1$ .

$a - 4 = {\sqrt{a + 5}}^{1} {\sqrt{a + 5}}^{1} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a - 4 = {\sqrt{a + 5}}^{1 + 1} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Add $1$ and $1$ .

$a - 4 = {\sqrt{a + 5}}^{2} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Rewrite ${\sqrt{a + 5}}^{2}$ as $a + 5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a + 5}$ as ${(a + 5)}^{\frac{1}{2}}$ .

$a - 4 = {({(a + 5)}^{\frac{1}{2}})}^{2} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

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

$a - 4 = {(a + 5)}^{\frac{1}{2} \cdot 2} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Combine $\frac{1}{2}$ and $2$ .

$a - 4 = {(a + 5)}^{\frac{2}{2}} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$a - 4 = {(a + 5)}^{\frac{2}{2}} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Rewrite the expression.

$a - 4 = {(a + 5)}^{1} + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Simplify.

$a - 4 = a + 5 + \sqrt{a + 5} \cdot - 1 - 1 \sqrt{a + 5} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{a + 5}$ .

$a - 4 = a + 5 - 1 \cdot \sqrt{a + 5} - 1 \sqrt{a + 5} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{a + 5}$ as $- \sqrt{a + 5}$ .

$a - 4 = a + 5 - \sqrt{a + 5} - 1 \sqrt{a + 5} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{a + 5}$ as $- \sqrt{a + 5}$ .

$a - 4 = a + 5 - \sqrt{a + 5} - \sqrt{a + 5} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$a - 4 = a + 5 - \sqrt{a + 5} - \sqrt{a + 5} + 1$

Add $5$ and $1$ .

$a - 4 = a + 6 - \sqrt{a + 5} - \sqrt{a + 5}$

Subtract $\sqrt{a + 5}$ from $- \sqrt{a + 5}$ .

$a - 4 = a + 6 - 2 \sqrt{a + 5}$

Step 4

Solve for $- 2 \sqrt{a + 5}$ .

Rewrite the equation as $a + 6 - 2 \sqrt{a + 5} = a - 4$ .

$a + 6 - 2 \sqrt{a + 5} = a - 4$

Move all terms not containing $- 2 \sqrt{a + 5}$ to the right side of the equation.

Subtract $a$ from both sides of the equation.

$6 - 2 \sqrt{a + 5} = a - 4 - a$

Subtract $6$ from both sides of the equation.

$- 2 \sqrt{a + 5} = a - 4 - a - 6$

Combine the opposite terms in $a - 4 - a - 6$ .

Subtract $a$ from $a$ .

$- 2 \sqrt{a + 5} = 0 - 4 - 6$

Subtract $4$ from $0$ .

$- 2 \sqrt{a + 5} = - 4 - 6$

Subtract $6$ from $- 4$ .

$- 2 \sqrt{a + 5} = - 10$

Step 5

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

${(- 2 \sqrt{a + 5})}^{2} = {(- 10)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a + 5}$ as ${(a + 5)}^{\frac{1}{2}}$ .

${(- 2 {(a + 5)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(a + 5)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(a + 5)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(a + 5)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Raise $- 2$ to the power of $2$ .

$4 {({(a + 5)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Multiply the exponents in ${({(a + 5)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(a + 5)}^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(a + 5)}^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Rewrite the expression.

$4 {(a + 5)}^{1} = {(- 10)}^{2}$

Simplify.

$4 (a + 5) = {(- 10)}^{2}$

Apply the distributive property.

$4 a + 4 \cdot 5 = {(- 10)}^{2}$

Multiply $4$ by $5$ .

$4 a + 20 = {(- 10)}^{2}$

Simplify the right side.

Raise $- 10$ to the power of $2$ .

$4 a + 20 = 100$

Step 7

Solve for $a$ .

Move all terms not containing $a$ to the right side of the equation.

Subtract $20$ from both sides of the equation.

$4 a = 100 - 20$

Subtract $20$ from $100$ .

$4 a = 80$

Divide each term in $4 a = 80$ by $4$ and simplify.

Divide each term in $4 a = 80$ by $4$ .

$\frac{4 a}{4} = \frac{80}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{80}{4}$

Divide $a$ by $1$ .

$a = \frac{80}{4}$

Simplify the right side.

Divide $80$ by $4$ .

$a = 20$

Solve for a square root of a-4+1 = square root of a+5

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