Solve for x square root of 4x-4 = square root of 9x+27

$\sqrt{4 x} - 4 = \sqrt{9 x + 27}$

Solve for $\sqrt{4 x}$ .

Simplify $\sqrt{9 x + 27}$ .

Rewrite.

$\sqrt{4 x} - 4 = 0 + 0 + \sqrt{9 x + 27}$

Simplify by adding zeros.

$\sqrt{4 x} - 4 = \sqrt{9 x + 27}$

Factor $9$ out of $9 x + 27$ .

Factor $9$ out of $9 x$ .

$\sqrt{4 x} - 4 = \sqrt{9 (x) + 27}$

Factor $9$ out of $27$ .

$\sqrt{4 x} - 4 = \sqrt{9 x + 9 \cdot 3}$

Factor $9$ out of $9 x + 9 \cdot 3$ .

$\sqrt{4 x} - 4 = \sqrt{9 (x + 3)}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Add $4$ to both sides of the equation.

$\sqrt{4 x} = 3 \sqrt{x + 3} + 4$

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

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

Simplify each side of the equation.

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

${({(4 x)}^{\frac{1}{2}})}^{2} = {(3 \sqrt{x + 3} + 4)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(4 x)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x + 3} + 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x + 3} + 4)}^{2}$

Rewrite the expression.

${(4 x)}^{1} = {(3 \sqrt{x + 3} + 4)}^{2}$

Simplify.

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

Simplify the right side.

Simplify ${(3 \sqrt{x + 3} + 4)}^{2}$ .

Rewrite ${(3 \sqrt{x + 3} + 4)}^{2}$ as $(3 \sqrt{x + 3} + 4) (3 \sqrt{x + 3} + 4)$ .

$4 x = (3 \sqrt{x + 3} + 4) (3 \sqrt{x + 3} + 4)$

Expand $(3 \sqrt{x + 3} + 4) (3 \sqrt{x + 3} + 4)$ using the FOIL Method.

Apply the distributive property.

$4 x = 3 \sqrt{x + 3} (3 \sqrt{x + 3} + 4) + 4 (3 \sqrt{x + 3} + 4)$

Apply the distributive property.

$4 x = 3 \sqrt{x + 3} (3 \sqrt{x + 3}) + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3} + 4)$

Apply the distributive property.

$4 x = 3 \sqrt{x + 3} (3 \sqrt{x + 3}) + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Multiply $3 \sqrt{x + 3} (3 \sqrt{x + 3})$ .

Multiply $3$ by $3$ .

$4 x = 9 \sqrt{x + 3} \sqrt{x + 3} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Raise $\sqrt{x + 3}$ to the power of $1$ .

$4 x = 9 ({\sqrt{x + 3}}^{1} \sqrt{x + 3}) + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Raise $\sqrt{x + 3}$ to the power of $1$ .

$4 x = 9 ({\sqrt{x + 3}}^{1} {\sqrt{x + 3}}^{1}) + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

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

$4 x = 9 {\sqrt{x + 3}}^{1 + 1} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Add $1$ and $1$ .

$4 x = 9 {\sqrt{x + 3}}^{2} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$4 x = 9 {({(x + 3)}^{\frac{1}{2}})}^{2} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

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

$4 x = 9 {(x + 3)}^{\frac{1}{2} \cdot 2} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

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

$4 x = 9 {(x + 3)}^{\frac{2}{2}} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x = 9 {(x + 3)}^{\frac{2}{2}} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Rewrite the expression.

$4 x = 9 {(x + 3)}^{1} + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Simplify.

$4 x = 9 (x + 3) + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Apply the distributive property.

$4 x = 9 x + 9 \cdot 3 + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Multiply $9$ by $3$ .

$4 x = 9 x + 27 + 3 \sqrt{x + 3} \cdot 4 + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Multiply $4$ by $3$ .

$4 x = 9 x + 27 + 12 \sqrt{x + 3} + 4 (3 \sqrt{x + 3}) + 4 \cdot 4$

Multiply $3$ by $4$ .

$4 x = 9 x + 27 + 12 \sqrt{x + 3} + 12 \sqrt{x + 3} + 4 \cdot 4$

Multiply $4$ by $4$ .

$4 x = 9 x + 27 + 12 \sqrt{x + 3} + 12 \sqrt{x + 3} + 16$

Add $27$ and $16$ .

$4 x = 9 x + 43 + 12 \sqrt{x + 3} + 12 \sqrt{x + 3}$

Add $12 \sqrt{x + 3}$ and $12 \sqrt{x + 3}$ .

$4 x = 9 x + 43 + 24 \sqrt{x + 3}$

Solve for $24 \sqrt{x + 3}$ .

Rewrite the equation as $9 x + 43 + 24 \sqrt{x + 3} = 4 x$ .

$9 x + 43 + 24 \sqrt{x + 3} = 4 x$

Move all terms not containing $24 \sqrt{x + 3}$ to the right side of the equation.

Subtract $9 x$ from both sides of the equation.

$43 + 24 \sqrt{x + 3} = 4 x - 9 x$

Subtract $43$ from both sides of the equation.

$24 \sqrt{x + 3} = 4 x - 9 x - 43$

Subtract $9 x$ from $4 x$ .

$24 \sqrt{x + 3} = - 5 x - 43$

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

${(24 \sqrt{x + 3})}^{2} = {(- 5 x - 43)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

${(24 {(x + 3)}^{\frac{1}{2}})}^{2} = {(- 5 x - 43)}^{2}$

Simplify the left side.

Simplify ${(24 {(x + 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $24 {(x + 3)}^{\frac{1}{2}}$ .

$24^{2} {({(x + 3)}^{\frac{1}{2}})}^{2} = {(- 5 x - 43)}^{2}$

Raise $24$ to the power of $2$ .

$576 {({(x + 3)}^{\frac{1}{2}})}^{2} = {(- 5 x - 43)}^{2}$

Multiply the exponents in ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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

$576 {(x + 3)}^{\frac{1}{2} \cdot 2} = {(- 5 x - 43)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$576 {(x + 3)}^{\frac{1}{2} \cdot 2} = {(- 5 x - 43)}^{2}$

Rewrite the expression.

$576 {(x + 3)}^{1} = {(- 5 x - 43)}^{2}$

Simplify.

$576 (x + 3) = {(- 5 x - 43)}^{2}$

Apply the distributive property.

$576 x + 576 \cdot 3 = {(- 5 x - 43)}^{2}$

Multiply $576$ by $3$ .

$576 x + 1728 = {(- 5 x - 43)}^{2}$

Simplify the right side.

Simplify ${(- 5 x - 43)}^{2}$ .

Rewrite ${(- 5 x - 43)}^{2}$ as $(- 5 x - 43) (- 5 x - 43)$ .

$576 x + 1728 = (- 5 x - 43) (- 5 x - 43)$

Expand $(- 5 x - 43) (- 5 x - 43)$ using the FOIL Method.

Apply the distributive property.

$576 x + 1728 = - 5 x (- 5 x - 43) - 43 (- 5 x - 43)$

Apply the distributive property.

$576 x + 1728 = - 5 x (- 5 x) - 5 x \cdot - 43 - 43 (- 5 x - 43)$

Apply the distributive property.

$576 x + 1728 = - 5 x (- 5 x) - 5 x \cdot - 43 - 43 (- 5 x) - 43 \cdot - 43$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$576 x + 1728 = - 5 \cdot - 5 x \cdot x - 5 x \cdot - 43 - 43 (- 5 x) - 43 \cdot - 43$

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

Move $x$ .

$576 x + 1728 = - 5 \cdot - 5 (x \cdot x) - 5 x \cdot - 43 - 43 (- 5 x) - 43 \cdot - 43$

Multiply $x$ by $x$ .

$576 x + 1728 = - 5 \cdot - 5 x^{2} - 5 x \cdot - 43 - 43 (- 5 x) - 43 \cdot - 43$

Multiply $- 5$ by $- 5$ .

$576 x + 1728 = 25 x^{2} - 5 x \cdot - 43 - 43 (- 5 x) - 43 \cdot - 43$

Multiply $- 43$ by $- 5$ .

$576 x + 1728 = 25 x^{2} + 215 x - 43 (- 5 x) - 43 \cdot - 43$

Multiply $- 5$ by $- 43$ .

$576 x + 1728 = 25 x^{2} + 215 x + 215 x - 43 \cdot - 43$

Multiply $- 43$ by $- 43$ .

$576 x + 1728 = 25 x^{2} + 215 x + 215 x + 1849$

Add $215 x$ and $215 x$ .

$576 x + 1728 = 25 x^{2} + 430 x + 1849$

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.

$25 x^{2} + 430 x + 1849 = 576 x + 1728$

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

Subtract $576 x$ from both sides of the equation.

$25 x^{2} + 430 x + 1849 - 576 x = 1728$

Subtract $576 x$ from $430 x$ .

$25 x^{2} - 146 x + 1849 = 1728$

Subtract $1728$ from both sides of the equation.

$25 x^{2} - 146 x + 1849 - 1728 = 0$

Subtract $1728$ from $1849$ .

$25 x^{2} - 146 x + 121 = 0$

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 $a \cdot c = 25 \cdot 121 = 3025$ and whose sum is $b = - 146$ .

Factor $- 146$ out of $- 146 x$ .

$25 x^{2} - 146 x + 121 = 0$

Rewrite $- 146$ as $- 25$ plus $- 121$

$25 x^{2} + (- 25 - 121) x + 121 = 0$

Apply the distributive property.

$25 x^{2} - 25 x - 121 x + 121 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(25 x^{2} - 25 x) - 121 x + 121 = 0$

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

$25 x (x - 1) - 121 (x - 1) = 0$

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

$(x - 1) (25 x - 121) = 0$

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

$x - 1 = 0$

$25 x - 121 = 0$

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 $25 x - 121$ equal to $0$ and solve for $x$ .

Set $25 x - 121$ equal to $0$ .

$25 x - 121 = 0$

Solve $25 x - 121 = 0$ for $x$ .

Add $121$ to both sides of the equation.

$25 x = 121$

Divide each term in $25 x = 121$ by $25$ and simplify.

Divide each term in $25 x = 121$ by $25$ .

$\frac{25 x}{25} = \frac{121}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 x}{25} = \frac{121}{25}$

Divide $x$ by $1$ .

$x = \frac{121}{25}$

The final solution is all the values that make $(x - 1) (25 x - 121) = 0$ true.

$x = 1, \frac{121}{25}$

Exclude the solutions that do not make $\sqrt{4 x} - 4 = \sqrt{9 x + 27}$ true.

$No solution$

Solve for x square root of 4x-4 = square root of 9x+27

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