Solve by Factoring 9- square root of x-9 = square root of 7x-6

$9 - \sqrt{x - 9} = \sqrt{7 x - 6}$

Step 1

Subtract $\sqrt{7 x - 6}$ from both sides of the equation.

$9 - \sqrt{x - 9} - \sqrt{7 x - 6} = 0$

Step 2

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

Subtract $9$ from both sides of the equation.

$- \sqrt{x - 9} - \sqrt{7 x - 6} = - 9$

Add $\sqrt{7 x - 6}$ to both sides of the equation.

$- \sqrt{x - 9} = - 9 + \sqrt{7 x - 6}$

Step 3

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

${(- \sqrt{x - 9})}^{2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Step 4

Simplify each side of the equation.

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

${(- {(x - 9)}^{\frac{1}{2}})}^{2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Simplify the left side.

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

Apply the product rule to $- {(x - 9)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x - 9)}^{\frac{1}{2}})}^{2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(x - 9)}^{\frac{1}{2}})}^{2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Multiply ${({(x - 9)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x - 9)}^{\frac{1}{2}})}^{2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

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

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

${(x - 9)}^{\frac{1}{2} \cdot 2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 9)}^{\frac{1}{2} \cdot 2} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Rewrite the expression.

${(x - 9)}^{1} = {(- 9 + \sqrt{7 x - 6})}^{2}$

Simplify.

$x - 9 = {(- 9 + \sqrt{7 x - 6})}^{2}$

Simplify the right side.

Simplify ${(- 9 + \sqrt{7 x - 6})}^{2}$ .

Rewrite ${(- 9 + \sqrt{7 x - 6})}^{2}$ as $(- 9 + \sqrt{7 x - 6}) (- 9 + \sqrt{7 x - 6})$ .

$x - 9 = (- 9 + \sqrt{7 x - 6}) (- 9 + \sqrt{7 x - 6})$

Expand $(- 9 + \sqrt{7 x - 6}) (- 9 + \sqrt{7 x - 6})$ using the FOIL Method.

Apply the distributive property.

$x - 9 = - 9 (- 9 + \sqrt{7 x - 6}) + \sqrt{7 x - 6} (- 9 + \sqrt{7 x - 6})$

Apply the distributive property.

$x - 9 = - 9 \cdot - 9 - 9 \sqrt{7 x - 6} + \sqrt{7 x - 6} (- 9 + \sqrt{7 x - 6})$

Apply the distributive property.

$x - 9 = - 9 \cdot - 9 - 9 \sqrt{7 x - 6} + \sqrt{7 x - 6} \cdot - 9 + \sqrt{7 x - 6} \sqrt{7 x - 6}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 9$ by $- 9$ .

$x - 9 = 81 - 9 \sqrt{7 x - 6} + \sqrt{7 x - 6} \cdot - 9 + \sqrt{7 x - 6} \sqrt{7 x - 6}$

Move $- 9$ to the left of $\sqrt{7 x - 6}$ .

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \cdot \sqrt{7 x - 6} + \sqrt{7 x - 6} \sqrt{7 x - 6}$

Multiply $\sqrt{7 x - 6} \sqrt{7 x - 6}$ .

Raise $\sqrt{7 x - 6}$ to the power of $1$ .

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {\sqrt{7 x - 6}}^{1} \sqrt{7 x - 6}$

Raise $\sqrt{7 x - 6}$ to the power of $1$ .

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {\sqrt{7 x - 6}}^{1} {\sqrt{7 x - 6}}^{1}$

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

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {\sqrt{7 x - 6}}^{1 + 1}$

Add $1$ and $1$ .

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {\sqrt{7 x - 6}}^{2}$

Rewrite ${\sqrt{7 x - 6}}^{2}$ as $7 x - 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x - 6}$ as ${(7 x - 6)}^{\frac{1}{2}}$ .

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {({(7 x - 6)}^{\frac{1}{2}})}^{2}$

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

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {(7 x - 6)}^{\frac{1}{2} \cdot 2}$

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

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {(7 x - 6)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {(7 x - 6)}^{\frac{2}{2}}$

Rewrite the expression.

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + {(7 x - 6)}^{1}$

Simplify.

$x - 9 = 81 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + 7 x - 6$

Subtract $6$ from $81$ .

$x - 9 = 75 - 9 \sqrt{7 x - 6} - 9 \sqrt{7 x - 6} + 7 x$

Subtract $9 \sqrt{7 x - 6}$ from $- 9 \sqrt{7 x - 6}$ .

$x - 9 = 75 - 18 \sqrt{7 x - 6} + 7 x$

Step 5

Solve for $- 18 \sqrt{7 x - 6}$ .

Rewrite the equation as $75 - 18 \sqrt{7 x - 6} + 7 x = x - 9$ .

$75 - 18 \sqrt{7 x - 6} + 7 x = x - 9$

Move all terms not containing $- 18 \sqrt{7 x - 6}$ to the right side of the equation.

Subtract $75$ from both sides of the equation.

$- 18 \sqrt{7 x - 6} + 7 x = x - 9 - 75$

Subtract $7 x$ from both sides of the equation.

$- 18 \sqrt{7 x - 6} = x - 9 - 75 - 7 x$

Subtract $7 x$ from $x$ .

$- 18 \sqrt{7 x - 6} = - 6 x - 9 - 75$

Subtract $75$ from $- 9$ .

$- 18 \sqrt{7 x - 6} = - 6 x - 84$

Step 6

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

${(- 18 \sqrt{7 x - 6})}^{2} = {(- 6 x - 84)}^{2}$

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x - 6}$ as ${(7 x - 6)}^{\frac{1}{2}}$ .

${(- 18 {(7 x - 6)}^{\frac{1}{2}})}^{2} = {(- 6 x - 84)}^{2}$

Simplify the left side.

Simplify ${(- 18 {(7 x - 6)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 18 {(7 x - 6)}^{\frac{1}{2}}$ .

${(- 18)}^{2} {({(7 x - 6)}^{\frac{1}{2}})}^{2} = {(- 6 x - 84)}^{2}$

Raise $- 18$ to the power of $2$ .

$324 {({(7 x - 6)}^{\frac{1}{2}})}^{2} = {(- 6 x - 84)}^{2}$

Multiply the exponents in ${({(7 x - 6)}^{\frac{1}{2}})}^{2}$ .

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

$324 {(7 x - 6)}^{\frac{1}{2} \cdot 2} = {(- 6 x - 84)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$324 {(7 x - 6)}^{\frac{1}{2} \cdot 2} = {(- 6 x - 84)}^{2}$

Rewrite the expression.

$324 {(7 x - 6)}^{1} = {(- 6 x - 84)}^{2}$

Simplify.

$324 (7 x - 6) = {(- 6 x - 84)}^{2}$

Apply the distributive property.

$324 (7 x) + 324 \cdot - 6 = {(- 6 x - 84)}^{2}$

Multiply.

Multiply $7$ by $324$ .

$2268 x + 324 \cdot - 6 = {(- 6 x - 84)}^{2}$

Multiply $324$ by $- 6$ .

$2268 x - 1944 = {(- 6 x - 84)}^{2}$

Simplify the right side.

Simplify ${(- 6 x - 84)}^{2}$ .

Rewrite ${(- 6 x - 84)}^{2}$ as $(- 6 x - 84) (- 6 x - 84)$ .

$2268 x - 1944 = (- 6 x - 84) (- 6 x - 84)$

Expand $(- 6 x - 84) (- 6 x - 84)$ using the FOIL Method.

Apply the distributive property.

$2268 x - 1944 = - 6 x (- 6 x - 84) - 84 (- 6 x - 84)$

Apply the distributive property.

$2268 x - 1944 = - 6 x (- 6 x) - 6 x \cdot - 84 - 84 (- 6 x - 84)$

Apply the distributive property.

$2268 x - 1944 = - 6 x (- 6 x) - 6 x \cdot - 84 - 84 (- 6 x) - 84 \cdot - 84$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2268 x - 1944 = - 6 \cdot - 6 x \cdot x - 6 x \cdot - 84 - 84 (- 6 x) - 84 \cdot - 84$

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

Move $x$ .

$2268 x - 1944 = - 6 \cdot - 6 (x \cdot x) - 6 x \cdot - 84 - 84 (- 6 x) - 84 \cdot - 84$

Multiply $x$ by $x$ .

$2268 x - 1944 = - 6 \cdot - 6 x^{2} - 6 x \cdot - 84 - 84 (- 6 x) - 84 \cdot - 84$

Multiply $- 6$ by $- 6$ .

$2268 x - 1944 = 36 x^{2} - 6 x \cdot - 84 - 84 (- 6 x) - 84 \cdot - 84$

Multiply $- 84$ by $- 6$ .

$2268 x - 1944 = 36 x^{2} + 504 x - 84 (- 6 x) - 84 \cdot - 84$

Multiply $- 6$ by $- 84$ .

$2268 x - 1944 = 36 x^{2} + 504 x + 504 x - 84 \cdot - 84$

Multiply $- 84$ by $- 84$ .

$2268 x - 1944 = 36 x^{2} + 504 x + 504 x + 7056$

Add $504 x$ and $504 x$ .

$2268 x - 1944 = 36 x^{2} + 1008 x + 7056$

Step 8

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.

$36 x^{2} + 1008 x + 7056 = 2268 x - 1944$

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

Subtract $2268 x$ from both sides of the equation.

$36 x^{2} + 1008 x + 7056 - 2268 x = - 1944$

Subtract $2268 x$ from $1008 x$ .

$36 x^{2} - 1260 x + 7056 = - 1944$

Add $1944$ to both sides of the equation.

$36 x^{2} - 1260 x + 7056 + 1944 = 0$

Add $7056$ and $1944$ .

$36 x^{2} - 1260 x + 9000 = 0$

Factor the left side of the equation.

Factor $36$ out of $36 x^{2} - 1260 x + 9000$ .

Factor $36$ out of $36 x^{2}$ .

$36 (x^{2}) - 1260 x + 9000 = 0$

Factor $36$ out of $- 1260 x$ .

$36 (x^{2}) + 36 (- 35 x) + 9000 = 0$

Factor $36$ out of $9000$ .

$36 x^{2} + 36 (- 35 x) + 36 \cdot 250 = 0$

Factor $36$ out of $36 x^{2} + 36 (- 35 x)$ .

$36 (x^{2} - 35 x) + 36 \cdot 250 = 0$

Factor $36$ out of $36 (x^{2} - 35 x) + 36 \cdot 250$ .

$36 (x^{2} - 35 x + 250) = 0$

Factor.

Factor $x^{2} - 35 x + 250$ 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 $250$ and whose sum is $- 35$ .

$- 25, - 10$

Write the factored form using these integers.

$36 ((x - 25) (x - 10)) = 0$

Remove unnecessary parentheses.

$36 (x - 25) (x - 10) = 0$

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

$x - 25 = 0$

$x - 10 = 0$

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

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

$x - 25 = 0$

Add $25$ to both sides of the equation.

$x = 25$

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

The final solution is all the values that make $36 (x - 25) (x - 10) = 0$ true.

$x = 25, 10$

Step 9

Exclude the solutions that do not make $9 - \sqrt{x - 9} - \sqrt{7 x - 6} = 0$ true.

$x = 10$

Solve by Factoring 9- square root of x-9 = square root of 7x-6

Related Questions

Support LCMGCF.com by disabling your adblocker.