Solve for x 2 square root of 9x-6=10-2 square root of x

$2 \sqrt{9 x} - 6 = 10 - 2 \sqrt{x}$

Step 1

Move all terms not containing $2 \sqrt{9 x}$ to the right side of the equation.

Add $6$ to both sides of the equation.

$2 \sqrt{9 x} = 10 - 2 \sqrt{x} + 6$

Add $10$ and $6$ .

$2 \sqrt{9 x} = 16 - 2 \sqrt{x}$

Step 2

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

${(2 \sqrt{9 x})}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Step 3

Simplify each side of the equation.

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

${(2 {(9 x)}^{\frac{1}{2}})}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Simplify the left side.

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

Simplify the expression.

Apply the product rule to $9 x$ .

${(2 (9^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = {(16 - 2 \sqrt{x})}^{2}$

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

${(2 ({(3^{2})}^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = {(16 - 2 \sqrt{x})}^{2}$

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

${(2 (3^{2 (\frac{1}{2})} x^{\frac{1}{2}}))}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 (3^{2 (\frac{1}{2})} x^{\frac{1}{2}}))}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Rewrite the expression.

${(2 (3^{1} x^{\frac{1}{2}}))}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Evaluate the exponent.

${(2 (3 x^{\frac{1}{2}}))}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Multiply $3$ by $2$ .

${(6 x^{\frac{1}{2}})}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Apply the product rule to $6 x^{\frac{1}{2}}$ .

$6^{2} {(x^{\frac{1}{2}})}^{2} = {(16 - 2 \sqrt{x})}^{2}$

Raise $6$ to the power of $2$ .

$36 {(x^{\frac{1}{2}})}^{2} = {(16 - 2 \sqrt{x})}^{2}$

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

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

$36 x^{\frac{1}{2} \cdot 2} = {(16 - 2 \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 x^{\frac{1}{2} \cdot 2} = {(16 - 2 \sqrt{x})}^{2}$

Rewrite the expression.

$36 x^{1} = {(16 - 2 \sqrt{x})}^{2}$

Simplify.

$36 x = {(16 - 2 \sqrt{x})}^{2}$

Simplify the right side.

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

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

$36 x = (16 - 2 \sqrt{x}) (16 - 2 \sqrt{x})$

Expand $(16 - 2 \sqrt{x}) (16 - 2 \sqrt{x})$ using the FOIL Method.

Apply the distributive property.

$36 x = 16 (16 - 2 \sqrt{x}) - 2 \sqrt{x} (16 - 2 \sqrt{x})$

Apply the distributive property.

$36 x = 16 \cdot 16 + 16 (- 2 \sqrt{x}) - 2 \sqrt{x} (16 - 2 \sqrt{x})$

Apply the distributive property.

$36 x = 16 \cdot 16 + 16 (- 2 \sqrt{x}) - 2 \sqrt{x} \cdot 16 - 2 \sqrt{x} (- 2 \sqrt{x})$

Simplify and combine like terms.

Simplify each term.

Multiply $16$ by $16$ .

$36 x = 256 + 16 (- 2 \sqrt{x}) - 2 \sqrt{x} \cdot 16 - 2 \sqrt{x} (- 2 \sqrt{x})$

Multiply $- 2$ by $16$ .

$36 x = 256 - 32 \sqrt{x} - 2 \sqrt{x} \cdot 16 - 2 \sqrt{x} (- 2 \sqrt{x})$

Multiply $16$ by $- 2$ .

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} - 2 \sqrt{x} (- 2 \sqrt{x})$

Multiply $- 2 \sqrt{x} (- 2 \sqrt{x})$ .

Multiply $- 2$ by $- 2$ .

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 \sqrt{x} \sqrt{x}$

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

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 ({\sqrt{x}}^{1} \sqrt{x})$

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

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})$

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

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 {\sqrt{x}}^{2}$

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

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

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 {(x^{\frac{1}{2}})}^{2}$

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

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 x^{\frac{1}{2} \cdot 2}$

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

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 x^{\frac{2}{2}}$

Rewrite the expression.

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 x^{1}$

Simplify.

$36 x = 256 - 32 \sqrt{x} - 32 \sqrt{x} + 4 x$

Subtract $32 \sqrt{x}$ from $- 32 \sqrt{x}$ .

$36 x = 256 - 64 \sqrt{x} + 4 x$

Step 4

Solve for $- 64 \sqrt{x}$ .

Rewrite the equation as $256 - 64 \sqrt{x} + 4 x = 36 x$ .

$256 - 64 \sqrt{x} + 4 x = 36 x$

Move all terms not containing $- 64 \sqrt{x}$ to the right side of the equation.

Subtract $256$ from both sides of the equation.

$- 64 \sqrt{x} + 4 x = 36 x - 256$

Subtract $4 x$ from both sides of the equation.

$- 64 \sqrt{x} = 36 x - 256 - 4 x$

Subtract $4 x$ from $36 x$ .

$- 64 \sqrt{x} = 32 x - 256$

Step 5

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

${(- 64 \sqrt{x})}^{2} = {(32 x - 256)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 64 x^{\frac{1}{2}})}^{2} = {(32 x - 256)}^{2}$

Simplify the left side.

Simplify ${(- 64 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 64 x^{\frac{1}{2}}$ .

${(- 64)}^{2} {(x^{\frac{1}{2}})}^{2} = {(32 x - 256)}^{2}$

Raise $- 64$ to the power of $2$ .

$4096 {(x^{\frac{1}{2}})}^{2} = {(32 x - 256)}^{2}$

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

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

$4096 x^{\frac{1}{2} \cdot 2} = {(32 x - 256)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4096 x^{\frac{1}{2} \cdot 2} = {(32 x - 256)}^{2}$

Rewrite the expression.

$4096 x^{1} = {(32 x - 256)}^{2}$

Simplify.

$4096 x = {(32 x - 256)}^{2}$

Simplify the right side.

Simplify ${(32 x - 256)}^{2}$ .

Rewrite ${(32 x - 256)}^{2}$ as $(32 x - 256) (32 x - 256)$ .

$4096 x = (32 x - 256) (32 x - 256)$

Expand $(32 x - 256) (32 x - 256)$ using the FOIL Method.

Apply the distributive property.

$4096 x = 32 x (32 x - 256) - 256 (32 x - 256)$

Apply the distributive property.

$4096 x = 32 x (32 x) + 32 x \cdot - 256 - 256 (32 x - 256)$

Apply the distributive property.

$4096 x = 32 x (32 x) + 32 x \cdot - 256 - 256 (32 x) - 256 \cdot - 256$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4096 x = 32 \cdot 32 x \cdot x + 32 x \cdot - 256 - 256 (32 x) - 256 \cdot - 256$

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

Move $x$ .

$4096 x = 32 \cdot 32 (x \cdot x) + 32 x \cdot - 256 - 256 (32 x) - 256 \cdot - 256$

Multiply $x$ by $x$ .

$4096 x = 32 \cdot 32 x^{2} + 32 x \cdot - 256 - 256 (32 x) - 256 \cdot - 256$

Multiply $32$ by $32$ .

$4096 x = 1024 x^{2} + 32 x \cdot - 256 - 256 (32 x) - 256 \cdot - 256$

Multiply $- 256$ by $32$ .

$4096 x = 1024 x^{2} - 8192 x - 256 (32 x) - 256 \cdot - 256$

Multiply $32$ by $- 256$ .

$4096 x = 1024 x^{2} - 8192 x - 8192 x - 256 \cdot - 256$

Multiply $- 256$ by $- 256$ .

$4096 x = 1024 x^{2} - 8192 x - 8192 x + 65536$

Subtract $8192 x$ from $- 8192 x$ .

$4096 x = 1024 x^{2} - 16384 x + 65536$

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.

$1024 x^{2} - 16384 x + 65536 = 4096 x$

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

Subtract $4096 x$ from both sides of the equation.

$1024 x^{2} - 16384 x + 65536 - 4096 x = 0$

Subtract $4096 x$ from $- 16384 x$ .

$1024 x^{2} - 20480 x + 65536 = 0$

Factor the left side of the equation.

Factor $1024$ out of $1024 x^{2} - 20480 x + 65536$ .

Factor $1024$ out of $1024 x^{2}$ .

$1024 (x^{2}) - 20480 x + 65536 = 0$

Factor $1024$ out of $- 20480 x$ .

$1024 (x^{2}) + 1024 (- 20 x) + 65536 = 0$

Factor $1024$ out of $65536$ .

$1024 x^{2} + 1024 (- 20 x) + 1024 \cdot 64 = 0$

Factor $1024$ out of $1024 x^{2} + 1024 (- 20 x)$ .

$1024 (x^{2} - 20 x) + 1024 \cdot 64 = 0$

Factor $1024$ out of $1024 (x^{2} - 20 x) + 1024 \cdot 64$ .

$1024 (x^{2} - 20 x + 64) = 0$

Factor.

Factor $x^{2} - 20 x + 64$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $64$ and whose sum is $- 20$ .

$- 16, - 4$

Write the factored form using these integers.

$1024 ((x - 16) (x - 4)) = 0$

Remove unnecessary parentheses.

$1024 (x - 16) (x - 4) = 0$

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

$x - 16 = 0$

$x - 4 = 0$

Set $x - 16$ equal to $0$ and solve for $x$ .

Set $x - 16$ equal to $0$ .

$x - 16 = 0$

Add $16$ to both sides of the equation.

$x = 16$

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 $1024 (x - 16) (x - 4) = 0$ true.

$x = 16, 4$

Step 8

Exclude the solutions that do not make $2 \sqrt{9 x} - 6 = 10 - 2 \sqrt{x}$ true.

$x = 4$

Solve for x 2 square root of 9x-6=10-2 square root of x

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