Solve for x square root of 5x+4=4+ square root of x+4

$\sqrt{5 x + 4} = 4 + \sqrt{x + 4}$

Step 1

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

${\sqrt{5 x + 4}}^{2} = {(4 + \sqrt{x + 4})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 4}$ as ${(5 x + 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(5 x + 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x + 4)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$5 x + 4 = {(4 + \sqrt{x + 4})}^{2}$

Simplify the right side.

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

Rewrite ${(4 + \sqrt{x + 4})}^{2}$ as $(4 + \sqrt{x + 4}) (4 + \sqrt{x + 4})$ .

$5 x + 4 = (4 + \sqrt{x + 4}) (4 + \sqrt{x + 4})$

Expand $(4 + \sqrt{x + 4}) (4 + \sqrt{x + 4})$ using the FOIL Method.

Apply the distributive property.

$5 x + 4 = 4 (4 + \sqrt{x + 4}) + \sqrt{x + 4} (4 + \sqrt{x + 4})$

Apply the distributive property.

$5 x + 4 = 4 \cdot 4 + 4 \sqrt{x + 4} + \sqrt{x + 4} (4 + \sqrt{x + 4})$

Apply the distributive property.

$5 x + 4 = 4 \cdot 4 + 4 \sqrt{x + 4} + \sqrt{x + 4} \cdot 4 + \sqrt{x + 4} \sqrt{x + 4}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$5 x + 4 = 16 + 4 \sqrt{x + 4} + \sqrt{x + 4} \cdot 4 + \sqrt{x + 4} \sqrt{x + 4}$

Move $4$ to the left of $\sqrt{x + 4}$ .

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \cdot \sqrt{x + 4} + \sqrt{x + 4} \sqrt{x + 4}$

Multiply $\sqrt{x + 4} \sqrt{x + 4}$ .

Raise $\sqrt{x + 4}$ to the power of $1$ .

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {\sqrt{x + 4}}^{1} \sqrt{x + 4}$

Raise $\sqrt{x + 4}$ to the power of $1$ .

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {\sqrt{x + 4}}^{1} {\sqrt{x + 4}}^{1}$

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

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {\sqrt{x + 4}}^{1 + 1}$

Add $1$ and $1$ .

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {\sqrt{x + 4}}^{2}$

Rewrite ${\sqrt{x + 4}}^{2}$ as $x + 4$ .

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

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {({(x + 4)}^{\frac{1}{2}})}^{2}$

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

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{\frac{1}{2} \cdot 2}$

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

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{\frac{2}{2}}$

Rewrite the expression.

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{1}$

Simplify.

$5 x + 4 = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + x + 4$

Add $16$ and $4$ .

$5 x + 4 = 20 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + x$

Add $4 \sqrt{x + 4}$ and $4 \sqrt{x + 4}$ .

$5 x + 4 = 20 + 8 \sqrt{x + 4} + x$

Step 3

Solve for $8 \sqrt{x + 4}$ .

Rewrite the equation as $20 + 8 \sqrt{x + 4} + x = 5 x + 4$ .

$20 + 8 \sqrt{x + 4} + x = 5 x + 4$

Move all terms not containing $8 \sqrt{x + 4}$ to the right side of the equation.

Subtract $20$ from both sides of the equation.

$8 \sqrt{x + 4} + x = 5 x + 4 - 20$

Subtract $x$ from both sides of the equation.

$8 \sqrt{x + 4} = 5 x + 4 - 20 - x$

Subtract $x$ from $5 x$ .

$8 \sqrt{x + 4} = 4 x + 4 - 20$

Subtract $20$ from $4$ .

$8 \sqrt{x + 4} = 4 x - 16$

Step 4

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

${(8 \sqrt{x + 4})}^{2} = {(4 x - 16)}^{2}$

Step 5

Simplify each side of the equation.

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

${(8 {(x + 4)}^{\frac{1}{2}})}^{2} = {(4 x - 16)}^{2}$

Simplify the left side.

Simplify ${(8 {(x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $8 {(x + 4)}^{\frac{1}{2}}$ .

$8^{2} {({(x + 4)}^{\frac{1}{2}})}^{2} = {(4 x - 16)}^{2}$

Raise $8$ to the power of $2$ .

$64 {({(x + 4)}^{\frac{1}{2}})}^{2} = {(4 x - 16)}^{2}$

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

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

$64 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(4 x - 16)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$64 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(4 x - 16)}^{2}$

Rewrite the expression.

$64 {(x + 4)}^{1} = {(4 x - 16)}^{2}$

Simplify.

$64 (x + 4) = {(4 x - 16)}^{2}$

Apply the distributive property.

$64 x + 64 \cdot 4 = {(4 x - 16)}^{2}$

Multiply $64$ by $4$ .

$64 x + 256 = {(4 x - 16)}^{2}$

Simplify the right side.

Simplify ${(4 x - 16)}^{2}$ .

Rewrite ${(4 x - 16)}^{2}$ as $(4 x - 16) (4 x - 16)$ .

$64 x + 256 = (4 x - 16) (4 x - 16)$

Expand $(4 x - 16) (4 x - 16)$ using the FOIL Method.

Apply the distributive property.

$64 x + 256 = 4 x (4 x - 16) - 16 (4 x - 16)$

Apply the distributive property.

$64 x + 256 = 4 x (4 x) + 4 x \cdot - 16 - 16 (4 x - 16)$

Apply the distributive property.

$64 x + 256 = 4 x (4 x) + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$64 x + 256 = 4 \cdot 4 x \cdot x + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

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

Move $x$ .

$64 x + 256 = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Multiply $x$ by $x$ .

$64 x + 256 = 4 \cdot 4 x^{2} + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Multiply $4$ by $4$ .

$64 x + 256 = 16 x^{2} + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Multiply $- 16$ by $4$ .

$64 x + 256 = 16 x^{2} - 64 x - 16 (4 x) - 16 \cdot - 16$

Multiply $4$ by $- 16$ .

$64 x + 256 = 16 x^{2} - 64 x - 64 x - 16 \cdot - 16$

Multiply $- 16$ by $- 16$ .

$64 x + 256 = 16 x^{2} - 64 x - 64 x + 256$

Subtract $64 x$ from $- 64 x$ .

$64 x + 256 = 16 x^{2} - 128 x + 256$

Step 6

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.

$16 x^{2} - 128 x + 256 = 64 x + 256$

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

Subtract $64 x$ from both sides of the equation.

$16 x^{2} - 128 x + 256 - 64 x = 256$

Subtract $64 x$ from $- 128 x$ .

$16 x^{2} - 192 x + 256 = 256$

Subtract $256$ from both sides of the equation.

$16 x^{2} - 192 x + 256 - 256 = 0$

Combine the opposite terms in $16 x^{2} - 192 x + 256 - 256$ .

Subtract $256$ from $256$ .

$16 x^{2} - 192 x + 0 = 0$

Add $16 x^{2} - 192 x$ and $0$ .

$16 x^{2} - 192 x = 0$

Factor $16 x$ out of $16 x^{2} - 192 x$ .

Factor $16 x$ out of $16 x^{2}$ .

$16 x (x) - 192 x = 0$

Factor $16 x$ out of $- 192 x$ .

$16 x (x) + 16 x (- 12) = 0$

Factor $16 x$ out of $16 x (x) + 16 x (- 12)$ .

$16 x (x - 12) = 0$

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

$x = 0$

$x - 12 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

The final solution is all the values that make $16 x (x - 12) = 0$ true.

$x = 0, 12$

Step 7

Exclude the solutions that do not make $\sqrt{5 x + 4} = 4 + \sqrt{x + 4}$ true.

$x = 12$

Solve for x square root of 5x+4=4+ square root of x+4

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