Solve for x square root of 4x+2 = square root of 9x-9

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

Step 1

Solve for $\sqrt{4 x}$ .

Simplify $\sqrt{9 x - 9}$ .

Rewrite.

$\sqrt{4 x} + 2 = 0 + 0 + \sqrt{9 x - 9}$

Simplify by adding zeros.

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

Factor $9$ out of $9 x - 9$ .

Factor $9$ out of $9 x$ .

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

Factor $9$ out of $- 9$ .

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

Factor $9$ out of $9 (x) + 9 (- 1)$ .

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

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

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

Pull terms out from under the radical.

$\sqrt{4 x} + 2 = 3 \sqrt{x - 1}$

Subtract $2$ from both sides of the equation.

$\sqrt{4 x} = 3 \sqrt{x - 1} - 2$

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(3 \sqrt{x - 1} - 2)}^{2}$ .

Rewrite ${(3 \sqrt{x - 1} - 2)}^{2}$ as $(3 \sqrt{x - 1} - 2) (3 \sqrt{x - 1} - 2)$ .

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

Expand $(3 \sqrt{x - 1} - 2) (3 \sqrt{x - 1} - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3 \sqrt{x - 1} (3 \sqrt{x - 1})$ .

Multiply $3$ by $3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $9$ by $- 1$ .

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

Multiply $- 2$ by $3$ .

$4 x = 9 x - 9 - 6 \sqrt{x - 1} - 2 (3 \sqrt{x - 1}) - 2 \cdot - 2$

Multiply $3$ by $- 2$ .

$4 x = 9 x - 9 - 6 \sqrt{x - 1} - 6 \sqrt{x - 1} - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$4 x = 9 x - 9 - 6 \sqrt{x - 1} - 6 \sqrt{x - 1} + 4$

Add $- 9$ and $4$ .

$4 x = 9 x - 5 - 6 \sqrt{x - 1} - 6 \sqrt{x - 1}$

Subtract $6 \sqrt{x - 1}$ from $- 6 \sqrt{x - 1}$ .

$4 x = 9 x - 5 - 12 \sqrt{x - 1}$

Step 4

Solve for $- 12 \sqrt{x - 1}$ .

Rewrite the equation as $9 x - 5 - 12 \sqrt{x - 1} = 4 x$ .

$9 x - 5 - 12 \sqrt{x - 1} = 4 x$

Move all terms not containing $- 12 \sqrt{x - 1}$ to the right side of the equation.

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

$- 5 - 12 \sqrt{x - 1} = 4 x - 9 x$

Add $5$ to both sides of the equation.

$- 12 \sqrt{x - 1} = 4 x - 9 x + 5$

Subtract $9 x$ from $4 x$ .

$- 12 \sqrt{x - 1} = - 5 x + 5$

Step 5

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

${(- 12 \sqrt{x - 1})}^{2} = {(- 5 x + 5)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 12 {(x - 1)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

Simplify the left side.

Simplify ${(- 12 {(x - 1)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 12 {(x - 1)}^{\frac{1}{2}}$ .

${(- 12)}^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

Raise $- 12$ to the power of $2$ .

$144 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

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

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

$144 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(- 5 x + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(- 5 x + 5)}^{2}$

Rewrite the expression.

$144 {(x - 1)}^{1} = {(- 5 x + 5)}^{2}$

Simplify.

$144 (x - 1) = {(- 5 x + 5)}^{2}$

Apply the distributive property.

$144 x + 144 \cdot - 1 = {(- 5 x + 5)}^{2}$

Multiply $144$ by $- 1$ .

$144 x - 144 = {(- 5 x + 5)}^{2}$

Simplify the right side.

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

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

$144 x - 144 = (- 5 x + 5) (- 5 x + 5)$

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

Apply the distributive property.

$144 x - 144 = - 5 x (- 5 x + 5) + 5 (- 5 x + 5)$

Apply the distributive property.

$144 x - 144 = - 5 x (- 5 x) - 5 x \cdot 5 + 5 (- 5 x + 5)$

Apply the distributive property.

$144 x - 144 = - 5 x (- 5 x) - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$144 x - 144 = - 5 \cdot - 5 x \cdot x - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

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

Move $x$ .

$144 x - 144 = - 5 \cdot - 5 (x \cdot x) - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Multiply $x$ by $x$ .

$144 x - 144 = - 5 \cdot - 5 x^{2} - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Multiply $- 5$ by $- 5$ .

$144 x - 144 = 25 x^{2} - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Multiply $5$ by $- 5$ .

$144 x - 144 = 25 x^{2} - 25 x + 5 (- 5 x) + 5 \cdot 5$

Multiply $- 5$ by $5$ .

$144 x - 144 = 25 x^{2} - 25 x - 25 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$144 x - 144 = 25 x^{2} - 25 x - 25 x + 25$

Subtract $25 x$ from $- 25 x$ .

$144 x - 144 = 25 x^{2} - 50 x + 25$

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.

$25 x^{2} - 50 x + 25 = 144 x - 144$

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

Subtract $144 x$ from both sides of the equation.

$25 x^{2} - 50 x + 25 - 144 x = - 144$

Subtract $144 x$ from $- 50 x$ .

$25 x^{2} - 194 x + 25 = - 144$

Add $144$ to both sides of the equation.

$25 x^{2} - 194 x + 25 + 144 = 0$

Add $25$ and $144$ .

$25 x^{2} - 194 x + 169 = 0$

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 $a \cdot c = 25 \cdot 169 = 4225$ and whose sum is $b = - 194$ .

Factor $- 194$ out of $- 194 x$ .

$25 x^{2} - 194 x + 169 = 0$

Rewrite $- 194$ as $- 25$ plus $- 169$

$25 x^{2} + (- 25 - 169) x + 169 = 0$

Apply the distributive property.

$25 x^{2} - 25 x - 169 x + 169 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(25 x^{2} - 25 x) - 169 x + 169 = 0$

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

$25 x (x - 1) - 169 (x - 1) = 0$

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

$(x - 1) (25 x - 169) = 0$

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

$x - 1 = 0$

$25 x - 169 = 0$

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 $25 x - 169$ equal to $0$ and solve for $x$ .

Set $25 x - 169$ equal to $0$ .

$25 x - 169 = 0$

Solve $25 x - 169 = 0$ for $x$ .

Add $169$ to both sides of the equation.

$25 x = 169$

Divide each term in $25 x = 169$ by $25$ and simplify.

Divide each term in $25 x = 169$ by $25$ .

$\frac{25 x}{25} = \frac{169}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 x}{25} = \frac{169}{25}$

Divide $x$ by $1$ .

$x = \frac{169}{25}$

The final solution is all the values that make $(x - 1) (25 x - 169) = 0$ true.

$x = 1, \frac{169}{25}$

Step 8

Exclude the solutions that do not make $\sqrt{4 x} + 2 = \sqrt{9 x - 9}$ true.

$x = \frac{169}{25}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{169}{25}$

Decimal Form:

$x = 6.76$

Mixed Number Form:

$x = 6 \frac{19}{25}$

Solve for x square root of 4x+2 = square root of 9x-9

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