Solve for x square root of 4x-2 = square root of x+15

$\sqrt{4 x} - 2 = \sqrt{x + 15}$

Add $2$ to both sides of the equation.

$\sqrt{4 x} = \sqrt{x + 15} + 2$

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

${\sqrt{4 x}}^{2} = {(\sqrt{x + 15} + 2)}^{2}$

Simplify each side of the equation.

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

${({(4 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 15} + 2)}^{2}$

Simplify the left side.

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

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

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

${(4 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 15} + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 15} + 2)}^{2}$

Rewrite the expression.

${(4 x)}^{1} = {(\sqrt{x + 15} + 2)}^{2}$

Simplify.

$4 x = {(\sqrt{x + 15} + 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 15} + 2)}^{2}$ .

Rewrite ${(\sqrt{x + 15} + 2)}^{2}$ as $(\sqrt{x + 15} + 2) (\sqrt{x + 15} + 2)$ .

$4 x = (\sqrt{x + 15} + 2) (\sqrt{x + 15} + 2)$

Expand $(\sqrt{x + 15} + 2) (\sqrt{x + 15} + 2)$ using the FOIL Method.

Apply the distributive property.

$4 x = \sqrt{x + 15} (\sqrt{x + 15} + 2) + 2 (\sqrt{x + 15} + 2)$

Apply the distributive property.

$4 x = \sqrt{x + 15} \sqrt{x + 15} + \sqrt{x + 15} \cdot 2 + 2 (\sqrt{x + 15} + 2)$

Apply the distributive property.

$4 x = \sqrt{x + 15} \sqrt{x + 15} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 15} \sqrt{x + 15}$ .

Raise $\sqrt{x + 15}$ to the power of $1$ .

$4 x = {\sqrt{x + 15}}^{1} \sqrt{x + 15} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Raise $\sqrt{x + 15}$ to the power of $1$ .

$4 x = {\sqrt{x + 15}}^{1} {\sqrt{x + 15}}^{1} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

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

$4 x = {\sqrt{x + 15}}^{1 + 1} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Add $1$ and $1$ .

$4 x = {\sqrt{x + 15}}^{2} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Rewrite ${\sqrt{x + 15}}^{2}$ as $x + 15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 15}$ as ${(x + 15)}^{\frac{1}{2}}$ .

$4 x = {({(x + 15)}^{\frac{1}{2}})}^{2} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

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

$4 x = {(x + 15)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

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

$4 x = {(x + 15)}^{\frac{2}{2}} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x = {(x + 15)}^{\frac{2}{2}} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Rewrite the expression.

$4 x = {(x + 15)}^{1} + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Simplify.

$4 x = x + 15 + \sqrt{x + 15} \cdot 2 + 2 \sqrt{x + 15} + 2 \cdot 2$

Move $2$ to the left of $\sqrt{x + 15}$ .

$4 x = x + 15 + 2 \cdot \sqrt{x + 15} + 2 \sqrt{x + 15} + 2 \cdot 2$

Multiply $2$ by $2$ .

$4 x = x + 15 + 2 \sqrt{x + 15} + 2 \sqrt{x + 15} + 4$

Add $15$ and $4$ .

$4 x = x + 19 + 2 \sqrt{x + 15} + 2 \sqrt{x + 15}$

Add $2 \sqrt{x + 15}$ and $2 \sqrt{x + 15}$ .

$4 x = x + 19 + 4 \sqrt{x + 15}$

Solve for $4 \sqrt{x + 15}$ .

Rewrite the equation as $x + 19 + 4 \sqrt{x + 15} = 4 x$ .

$x + 19 + 4 \sqrt{x + 15} = 4 x$

Move all terms not containing $4 \sqrt{x + 15}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$19 + 4 \sqrt{x + 15} = 4 x - x$

Subtract $19$ from both sides of the equation.

$4 \sqrt{x + 15} = 4 x - x - 19$

Subtract $x$ from $4 x$ .

$4 \sqrt{x + 15} = 3 x - 19$

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

${(4 \sqrt{x + 15})}^{2} = {(3 x - 19)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 15}$ as ${(x + 15)}^{\frac{1}{2}}$ .

${(4 {(x + 15)}^{\frac{1}{2}})}^{2} = {(3 x - 19)}^{2}$

Simplify the left side.

Simplify ${(4 {(x + 15)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(x + 15)}^{\frac{1}{2}}$ .

$4^{2} {({(x + 15)}^{\frac{1}{2}})}^{2} = {(3 x - 19)}^{2}$

Raise $4$ to the power of $2$ .

$16 {({(x + 15)}^{\frac{1}{2}})}^{2} = {(3 x - 19)}^{2}$

Multiply the exponents in ${({(x + 15)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(x + 15)}^{\frac{1}{2} \cdot 2} = {(3 x - 19)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x + 15)}^{\frac{1}{2} \cdot 2} = {(3 x - 19)}^{2}$

Rewrite the expression.

$16 {(x + 15)}^{1} = {(3 x - 19)}^{2}$

Simplify.

$16 (x + 15) = {(3 x - 19)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot 15 = {(3 x - 19)}^{2}$

Multiply $16$ by $15$ .

$16 x + 240 = {(3 x - 19)}^{2}$

Simplify the right side.

Simplify ${(3 x - 19)}^{2}$ .

Rewrite ${(3 x - 19)}^{2}$ as $(3 x - 19) (3 x - 19)$ .

$16 x + 240 = (3 x - 19) (3 x - 19)$

Expand $(3 x - 19) (3 x - 19)$ using the FOIL Method.

Apply the distributive property.

$16 x + 240 = 3 x (3 x - 19) - 19 (3 x - 19)$

Apply the distributive property.

$16 x + 240 = 3 x (3 x) + 3 x \cdot - 19 - 19 (3 x - 19)$

Apply the distributive property.

$16 x + 240 = 3 x (3 x) + 3 x \cdot - 19 - 19 (3 x) - 19 \cdot - 19$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$16 x + 240 = 3 \cdot 3 x \cdot x + 3 x \cdot - 19 - 19 (3 x) - 19 \cdot - 19$

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

Move $x$ .

$16 x + 240 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 19 - 19 (3 x) - 19 \cdot - 19$

Multiply $x$ by $x$ .

$16 x + 240 = 3 \cdot 3 x^{2} + 3 x \cdot - 19 - 19 (3 x) - 19 \cdot - 19$

Multiply $3$ by $3$ .

$16 x + 240 = 9 x^{2} + 3 x \cdot - 19 - 19 (3 x) - 19 \cdot - 19$

Multiply $- 19$ by $3$ .

$16 x + 240 = 9 x^{2} - 57 x - 19 (3 x) - 19 \cdot - 19$

Multiply $3$ by $- 19$ .

$16 x + 240 = 9 x^{2} - 57 x - 57 x - 19 \cdot - 19$

Multiply $- 19$ by $- 19$ .

$16 x + 240 = 9 x^{2} - 57 x - 57 x + 361$

Subtract $57 x$ from $- 57 x$ .

$16 x + 240 = 9 x^{2} - 114 x + 361$

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.

$9 x^{2} - 114 x + 361 = 16 x + 240$

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

Subtract $16 x$ from both sides of the equation.

$9 x^{2} - 114 x + 361 - 16 x = 240$

Subtract $16 x$ from $- 114 x$ .

$9 x^{2} - 130 x + 361 = 240$

Subtract $240$ from both sides of the equation.

$9 x^{2} - 130 x + 361 - 240 = 0$

Subtract $240$ from $361$ .

$9 x^{2} - 130 x + 121 = 0$

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 $a \cdot c = 9 \cdot 121 = 1089$ and whose sum is $b = - 130$ .

Factor $- 130$ out of $- 130 x$ .

$9 x^{2} - 130 x + 121 = 0$

Rewrite $- 130$ as $- 9$ plus $- 121$

$9 x^{2} + (- 9 - 121) x + 121 = 0$

Apply the distributive property.

$9 x^{2} - 9 x - 121 x + 121 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 x^{2} - 9 x) - 121 x + 121 = 0$

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

$9 x (x - 1) - 121 (x - 1) = 0$

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

$(x - 1) (9 x - 121) = 0$

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

$x - 1 = 0$

$9 x - 121 = 0$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $9 x - 121$ equal to $0$ and solve for $x$ .

Set $9 x - 121$ equal to $0$ .

$9 x - 121 = 0$

Solve $9 x - 121 = 0$ for $x$ .

Add $121$ to both sides of the equation.

$9 x = 121$

Divide each term in $9 x = 121$ by $9$ and simplify.

Divide each term in $9 x = 121$ by $9$ .

$\frac{9 x}{9} = \frac{121}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{121}{9}$

Divide $x$ by $1$ .

$x = \frac{121}{9}$

The final solution is all the values that make $(x - 1) (9 x - 121) = 0$ true.

$x = 1, \frac{121}{9}$

Exclude the solutions that do not make $\sqrt{4 x} - 2 = \sqrt{x + 15}$ true.

$x = \frac{121}{9}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{121}{9}$

Decimal Form:

$x = 13.\overline{4}$

Mixed Number Form:

$x = 13 \frac{4}{9}$

Solve for x square root of 4x-2 = square root of x+15

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