Solve for x square root of 9x+4 = square root of x+30

$\sqrt{9 x} + 4 = \sqrt{x + 30}$

Subtract $4$ from both sides of the equation.

$\sqrt{9 x} = \sqrt{x + 30} - 4$

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

${\sqrt{9 x}}^{2} = {(\sqrt{x + 30} - 4)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x}$ as ${(9 x)}^{\frac{1}{2}}$ .

${({(9 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 30} - 4)}^{2}$

Simplify the left side.

Simplify ${({(9 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(9 x)}^{\frac{1}{2}})}^{2}$ .

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

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 30} - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 30} - 4)}^{2}$

Rewrite the expression.

${(9 x)}^{1} = {(\sqrt{x + 30} - 4)}^{2}$

Simplify.

$9 x = {(\sqrt{x + 30} - 4)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 30} - 4)}^{2}$ .

Rewrite ${(\sqrt{x + 30} - 4)}^{2}$ as $(\sqrt{x + 30} - 4) (\sqrt{x + 30} - 4)$ .

$9 x = (\sqrt{x + 30} - 4) (\sqrt{x + 30} - 4)$

Expand $(\sqrt{x + 30} - 4) (\sqrt{x + 30} - 4)$ using the FOIL Method.

Apply the distributive property.

$9 x = \sqrt{x + 30} (\sqrt{x + 30} - 4) - 4 (\sqrt{x + 30} - 4)$

Apply the distributive property.

$9 x = \sqrt{x + 30} \sqrt{x + 30} + \sqrt{x + 30} \cdot - 4 - 4 (\sqrt{x + 30} - 4)$

Apply the distributive property.

$9 x = \sqrt{x + 30} \sqrt{x + 30} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 30} \sqrt{x + 30}$ .

Raise $\sqrt{x + 30}$ to the power of $1$ .

$9 x = {\sqrt{x + 30}}^{1} \sqrt{x + 30} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Raise $\sqrt{x + 30}$ to the power of $1$ .

$9 x = {\sqrt{x + 30}}^{1} {\sqrt{x + 30}}^{1} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

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

$9 x = {\sqrt{x + 30}}^{1 + 1} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Add $1$ and $1$ .

$9 x = {\sqrt{x + 30}}^{2} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Rewrite ${\sqrt{x + 30}}^{2}$ as $x + 30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 30}$ as ${(x + 30)}^{\frac{1}{2}}$ .

$9 x = {({(x + 30)}^{\frac{1}{2}})}^{2} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

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

$9 x = {(x + 30)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

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

$9 x = {(x + 30)}^{\frac{2}{2}} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x = {(x + 30)}^{\frac{2}{2}} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Rewrite the expression.

$9 x = {(x + 30)}^{1} + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Simplify.

$9 x = x + 30 + \sqrt{x + 30} \cdot - 4 - 4 \sqrt{x + 30} - 4 \cdot - 4$

Move $- 4$ to the left of $\sqrt{x + 30}$ .

$9 x = x + 30 - 4 \cdot \sqrt{x + 30} - 4 \sqrt{x + 30} - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$9 x = x + 30 - 4 \sqrt{x + 30} - 4 \sqrt{x + 30} + 16$

Add $30$ and $16$ .

$9 x = x + 46 - 4 \sqrt{x + 30} - 4 \sqrt{x + 30}$

Subtract $4 \sqrt{x + 30}$ from $- 4 \sqrt{x + 30}$ .

$9 x = x + 46 - 8 \sqrt{x + 30}$

Solve for $- 8 \sqrt{x + 30}$ .

Rewrite the equation as $x + 46 - 8 \sqrt{x + 30} = 9 x$ .

$x + 46 - 8 \sqrt{x + 30} = 9 x$

Move all terms not containing $- 8 \sqrt{x + 30}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$46 - 8 \sqrt{x + 30} = 9 x - x$

Subtract $46$ from both sides of the equation.

$- 8 \sqrt{x + 30} = 9 x - x - 46$

Subtract $x$ from $9 x$ .

$- 8 \sqrt{x + 30} = 8 x - 46$

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

${(- 8 \sqrt{x + 30})}^{2} = {(8 x - 46)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 30}$ as ${(x + 30)}^{\frac{1}{2}}$ .

${(- 8 {(x + 30)}^{\frac{1}{2}})}^{2} = {(8 x - 46)}^{2}$

Simplify the left side.

Simplify ${(- 8 {(x + 30)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 8 {(x + 30)}^{\frac{1}{2}}$ .

${(- 8)}^{2} {({(x + 30)}^{\frac{1}{2}})}^{2} = {(8 x - 46)}^{2}$

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

$64 {({(x + 30)}^{\frac{1}{2}})}^{2} = {(8 x - 46)}^{2}$

Multiply the exponents in ${({(x + 30)}^{\frac{1}{2}})}^{2}$ .

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

$64 {(x + 30)}^{\frac{1}{2} \cdot 2} = {(8 x - 46)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$64 {(x + 30)}^{\frac{1}{2} \cdot 2} = {(8 x - 46)}^{2}$

Rewrite the expression.

$64 {(x + 30)}^{1} = {(8 x - 46)}^{2}$

Simplify.

$64 (x + 30) = {(8 x - 46)}^{2}$

Apply the distributive property.

$64 x + 64 \cdot 30 = {(8 x - 46)}^{2}$

Multiply $64$ by $30$ .

$64 x + 1920 = {(8 x - 46)}^{2}$

Simplify the right side.

Simplify ${(8 x - 46)}^{2}$ .

Rewrite ${(8 x - 46)}^{2}$ as $(8 x - 46) (8 x - 46)$ .

$64 x + 1920 = (8 x - 46) (8 x - 46)$

Expand $(8 x - 46) (8 x - 46)$ using the FOIL Method.

Apply the distributive property.

$64 x + 1920 = 8 x (8 x - 46) - 46 (8 x - 46)$

Apply the distributive property.

$64 x + 1920 = 8 x (8 x) + 8 x \cdot - 46 - 46 (8 x - 46)$

Apply the distributive property.

$64 x + 1920 = 8 x (8 x) + 8 x \cdot - 46 - 46 (8 x) - 46 \cdot - 46$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$64 x + 1920 = 8 \cdot 8 x \cdot x + 8 x \cdot - 46 - 46 (8 x) - 46 \cdot - 46$

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

Move $x$ .

$64 x + 1920 = 8 \cdot 8 (x \cdot x) + 8 x \cdot - 46 - 46 (8 x) - 46 \cdot - 46$

Multiply $x$ by $x$ .

$64 x + 1920 = 8 \cdot 8 x^{2} + 8 x \cdot - 46 - 46 (8 x) - 46 \cdot - 46$

Multiply $8$ by $8$ .

$64 x + 1920 = 64 x^{2} + 8 x \cdot - 46 - 46 (8 x) - 46 \cdot - 46$

Multiply $- 46$ by $8$ .

$64 x + 1920 = 64 x^{2} - 368 x - 46 (8 x) - 46 \cdot - 46$

Multiply $8$ by $- 46$ .

$64 x + 1920 = 64 x^{2} - 368 x - 368 x - 46 \cdot - 46$

Multiply $- 46$ by $- 46$ .

$64 x + 1920 = 64 x^{2} - 368 x - 368 x + 2116$

Subtract $368 x$ from $- 368 x$ .

$64 x + 1920 = 64 x^{2} - 736 x + 2116$

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.

$64 x^{2} - 736 x + 2116 = 64 x + 1920$

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

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

$64 x^{2} - 736 x + 2116 - 64 x = 1920$

Subtract $64 x$ from $- 736 x$ .

$64 x^{2} - 800 x + 2116 = 1920$

Subtract $1920$ from both sides of the equation.

$64 x^{2} - 800 x + 2116 - 1920 = 0$

Subtract $1920$ from $2116$ .

$64 x^{2} - 800 x + 196 = 0$

Factor the left side of the equation.

Factor $4$ out of $64 x^{2} - 800 x + 196$ .

Factor $4$ out of $64 x^{2}$ .

$4 (16 x^{2}) - 800 x + 196 = 0$

Factor $4$ out of $- 800 x$ .

$4 (16 x^{2}) + 4 (- 200 x) + 196 = 0$

Factor $4$ out of $196$ .

$4 (16 x^{2}) + 4 (- 200 x) + 4 (49) = 0$

Factor $4$ out of $4 (16 x^{2}) + 4 (- 200 x)$ .

$4 (16 x^{2} - 200 x) + 4 (49) = 0$

Factor $4$ out of $4 (16 x^{2} - 200 x) + 4 (49)$ .

$4 (16 x^{2} - 200 x + 49) = 0$

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 $a \cdot c = 16 \cdot 49 = 784$ and whose sum is $b = - 200$ .

Factor $- 200$ out of $- 200 x$ .

$4 (16 x^{2} - 200 x + 49) = 0$

Rewrite $- 200$ as $- 4$ plus $- 196$

$4 (16 x^{2} + (- 4 - 196) x + 49) = 0$

Apply the distributive property.

$4 (16 x^{2} - 4 x - 196 x + 49) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((16 x^{2} - 4 x) - 196 x + 49) = 0$

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

$4 (4 x (4 x - 1) - 49 (4 x - 1)) = 0$

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

$4 ((4 x - 1) (4 x - 49)) = 0$

Remove unnecessary parentheses.

$4 (4 x - 1) (4 x - 49) = 0$

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

$4 x - 1 = 0$

$4 x - 49 = 0$

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 $4 x - 49$ equal to $0$ and solve for $x$ .

Set $4 x - 49$ equal to $0$ .

$4 x - 49 = 0$

Solve $4 x - 49 = 0$ for $x$ .

Add $49$ to both sides of the equation.

$4 x = 49$

Divide each term in $4 x = 49$ by $4$ and simplify.

Divide each term in $4 x = 49$ by $4$ .

$\frac{4 x}{4} = \frac{49}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{49}{4}$

Divide $x$ by $1$ .

$x = \frac{49}{4}$

The final solution is all the values that make $4 (4 x - 1) (4 x - 49) = 0$ true.

$x = \frac{1}{4}, \frac{49}{4}$

Exclude the solutions that do not make $\sqrt{9 x} + 4 = \sqrt{x + 30}$ true.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.25$

Solve for x square root of 9x+4 = square root of x+30

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