Solve for x square root of x-5 = square root of 5x+29

$\sqrt{x} - 5 = \sqrt{5 x + 29}$

Step 1

Add $5$ to both sides of the equation.

$\sqrt{x} = \sqrt{5 x + 29} + 5$

Step 2

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

${\sqrt{x}}^{2} = {(\sqrt{5 x + 29} + 5)}^{2}$

Step 3

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{5 x + 29} + 5)}^{2}$

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{5 x + 29} + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{5 x + 29} + 5)}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{5 x + 29} + 5)}^{2}$

Simplify.

$x = {(\sqrt{5 x + 29} + 5)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{5 x + 29} + 5)}^{2}$ .

Rewrite ${(\sqrt{5 x + 29} + 5)}^{2}$ as $(\sqrt{5 x + 29} + 5) (\sqrt{5 x + 29} + 5)$ .

$x = (\sqrt{5 x + 29} + 5) (\sqrt{5 x + 29} + 5)$

Expand $(\sqrt{5 x + 29} + 5) (\sqrt{5 x + 29} + 5)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{5 x + 29} (\sqrt{5 x + 29} + 5) + 5 (\sqrt{5 x + 29} + 5)$

Apply the distributive property.

$x = \sqrt{5 x + 29} \sqrt{5 x + 29} + \sqrt{5 x + 29} \cdot 5 + 5 (\sqrt{5 x + 29} + 5)$

Apply the distributive property.

$x = \sqrt{5 x + 29} \sqrt{5 x + 29} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{5 x + 29} \sqrt{5 x + 29}$ .

Raise $\sqrt{5 x + 29}$ to the power of $1$ .

$x = {\sqrt{5 x + 29}}^{1} \sqrt{5 x + 29} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Raise $\sqrt{5 x + 29}$ to the power of $1$ .

$x = {\sqrt{5 x + 29}}^{1} {\sqrt{5 x + 29}}^{1} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

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

$x = {\sqrt{5 x + 29}}^{1 + 1} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Add $1$ and $1$ .

$x = {\sqrt{5 x + 29}}^{2} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Rewrite ${\sqrt{5 x + 29}}^{2}$ as $5 x + 29$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 29}$ as ${(5 x + 29)}^{\frac{1}{2}}$ .

$x = {({(5 x + 29)}^{\frac{1}{2}})}^{2} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

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

$x = {(5 x + 29)}^{\frac{1}{2} \cdot 2} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

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

$x = {(5 x + 29)}^{\frac{2}{2}} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(5 x + 29)}^{\frac{2}{2}} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Rewrite the expression.

$x = {(5 x + 29)}^{1} + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Simplify.

$x = 5 x + 29 + \sqrt{5 x + 29} \cdot 5 + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Move $5$ to the left of $\sqrt{5 x + 29}$ .

$x = 5 x + 29 + 5 \cdot \sqrt{5 x + 29} + 5 \sqrt{5 x + 29} + 5 \cdot 5$

Multiply $5$ by $5$ .

$x = 5 x + 29 + 5 \sqrt{5 x + 29} + 5 \sqrt{5 x + 29} + 25$

Add $29$ and $25$ .

$x = 5 x + 54 + 5 \sqrt{5 x + 29} + 5 \sqrt{5 x + 29}$

Add $5 \sqrt{5 x + 29}$ and $5 \sqrt{5 x + 29}$ .

$x = 5 x + 54 + 10 \sqrt{5 x + 29}$

Step 4

Solve for $10 \sqrt{5 x + 29}$ .

Rewrite the equation as $5 x + 54 + 10 \sqrt{5 x + 29} = x$ .

$5 x + 54 + 10 \sqrt{5 x + 29} = x$

Move all terms not containing $10 \sqrt{5 x + 29}$ to the right side of the equation.

Subtract $5 x$ from both sides of the equation.

$54 + 10 \sqrt{5 x + 29} = x - 5 x$

Subtract $54$ from both sides of the equation.

$10 \sqrt{5 x + 29} = x - 5 x - 54$

Subtract $5 x$ from $x$ .

$10 \sqrt{5 x + 29} = - 4 x - 54$

Step 5

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

${(10 \sqrt{5 x + 29})}^{2} = {(- 4 x - 54)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 29}$ as ${(5 x + 29)}^{\frac{1}{2}}$ .

${(10 {(5 x + 29)}^{\frac{1}{2}})}^{2} = {(- 4 x - 54)}^{2}$

Simplify the left side.

Simplify ${(10 {(5 x + 29)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $10 {(5 x + 29)}^{\frac{1}{2}}$ .

$10^{2} {({(5 x + 29)}^{\frac{1}{2}})}^{2} = {(- 4 x - 54)}^{2}$

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

$100 {({(5 x + 29)}^{\frac{1}{2}})}^{2} = {(- 4 x - 54)}^{2}$

Multiply the exponents in ${({(5 x + 29)}^{\frac{1}{2}})}^{2}$ .

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

$100 {(5 x + 29)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 54)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(5 x + 29)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 54)}^{2}$

Rewrite the expression.

$100 {(5 x + 29)}^{1} = {(- 4 x - 54)}^{2}$

Simplify.

$100 (5 x + 29) = {(- 4 x - 54)}^{2}$

Apply the distributive property.

$100 (5 x) + 100 \cdot 29 = {(- 4 x - 54)}^{2}$

Multiply.

Multiply $5$ by $100$ .

$500 x + 100 \cdot 29 = {(- 4 x - 54)}^{2}$

Multiply $100$ by $29$ .

$500 x + 2900 = {(- 4 x - 54)}^{2}$

Simplify the right side.

Simplify ${(- 4 x - 54)}^{2}$ .

Rewrite ${(- 4 x - 54)}^{2}$ as $(- 4 x - 54) (- 4 x - 54)$ .

$500 x + 2900 = (- 4 x - 54) (- 4 x - 54)$

Expand $(- 4 x - 54) (- 4 x - 54)$ using the FOIL Method.

Apply the distributive property.

$500 x + 2900 = - 4 x (- 4 x - 54) - 54 (- 4 x - 54)$

Apply the distributive property.

$500 x + 2900 = - 4 x (- 4 x) - 4 x \cdot - 54 - 54 (- 4 x - 54)$

Apply the distributive property.

$500 x + 2900 = - 4 x (- 4 x) - 4 x \cdot - 54 - 54 (- 4 x) - 54 \cdot - 54$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$500 x + 2900 = - 4 \cdot - 4 x \cdot x - 4 x \cdot - 54 - 54 (- 4 x) - 54 \cdot - 54$

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

Move $x$ .

$500 x + 2900 = - 4 \cdot - 4 (x \cdot x) - 4 x \cdot - 54 - 54 (- 4 x) - 54 \cdot - 54$

Multiply $x$ by $x$ .

$500 x + 2900 = - 4 \cdot - 4 x^{2} - 4 x \cdot - 54 - 54 (- 4 x) - 54 \cdot - 54$

Multiply $- 4$ by $- 4$ .

$500 x + 2900 = 16 x^{2} - 4 x \cdot - 54 - 54 (- 4 x) - 54 \cdot - 54$

Multiply $- 54$ by $- 4$ .

$500 x + 2900 = 16 x^{2} + 216 x - 54 (- 4 x) - 54 \cdot - 54$

Multiply $- 4$ by $- 54$ .

$500 x + 2900 = 16 x^{2} + 216 x + 216 x - 54 \cdot - 54$

Multiply $- 54$ by $- 54$ .

$500 x + 2900 = 16 x^{2} + 216 x + 216 x + 2916$

Add $216 x$ and $216 x$ .

$500 x + 2900 = 16 x^{2} + 432 x + 2916$

Step 7

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.

$16 x^{2} + 432 x + 2916 = 500 x + 2900$

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

Subtract $500 x$ from both sides of the equation.

$16 x^{2} + 432 x + 2916 - 500 x = 2900$

Subtract $500 x$ from $432 x$ .

$16 x^{2} - 68 x + 2916 = 2900$

Subtract $2900$ from both sides of the equation.

$16 x^{2} - 68 x + 2916 - 2900 = 0$

Subtract $2900$ from $2916$ .

$16 x^{2} - 68 x + 16 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} - 68 x + 16$ .

Factor $4$ out of $16 x^{2}$ .

$4 (4 x^{2}) - 68 x + 16 = 0$

Factor $4$ out of $- 68 x$ .

$4 (4 x^{2}) + 4 (- 17 x) + 16 = 0$

Factor $4$ out of $16$ .

$4 (4 x^{2}) + 4 (- 17 x) + 4 (4) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (- 17 x)$ .

$4 (4 x^{2} - 17 x) + 4 (4) = 0$

Factor $4$ out of $4 (4 x^{2} - 17 x) + 4 (4)$ .

$4 (4 x^{2} - 17 x + 4) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 4 = 16$ and whose sum is $b = - 17$ .

Factor $- 17$ out of $- 17 x$ .

$4 (4 x^{2} - 17 x + 4) = 0$

Rewrite $- 17$ as $- 1$ plus $- 16$

$4 (4 x^{2} + (- 1 - 16) x + 4) = 0$

Apply the distributive property.

$4 (4 x^{2} - 1 x - 16 x + 4) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} - 1 x) - 16 x + 4) = 0$

Factor out the greatest common factor (GCF) from each group.

$4 (x (4 x - 1) - 4 (4 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$4 ((4 x - 1) (x - 4)) = 0$

Remove unnecessary parentheses.

$4 (4 x - 1) (x - 4) = 0$

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

$4 x - 1 = 0$

$x - 4 = 0$

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $4 (4 x - 1) (x - 4) = 0$ true.

$x = \frac{1}{4}, 4$

Step 8

Exclude the solutions that do not make $\sqrt{x} - 5 = \sqrt{5 x + 29}$ true.

$No solution$

Solve for x square root of x-5 = square root of 5x+29

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