Solve for x x- square root of 169-x^2=17

$x - \sqrt{169 - x^{2}} = 17$

Subtract $x$ from both sides of the equation.

$- \sqrt{169 - x^{2}} = 17 - x$

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

${(- \sqrt{169 - x^{2}})}^{2} = {(17 - x)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{169 - x^{2}}$ as ${(169 - x^{2})}^{\frac{1}{2}}$ .

${(- {(169 - x^{2})}^{\frac{1}{2}})}^{2} = {(17 - x)}^{2}$

Simplify the left side.

Simplify ${(- {(169 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(169 - x^{2})}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(169 - x^{2})}^{\frac{1}{2}})}^{2} = {(17 - x)}^{2}$

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

$1 {({(169 - x^{2})}^{\frac{1}{2}})}^{2} = {(17 - x)}^{2}$

Multiply ${({(169 - x^{2})}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(169 - x^{2})}^{\frac{1}{2}})}^{2} = {(17 - x)}^{2}$

Multiply the exponents in ${({(169 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(169 - x^{2})}^{\frac{1}{2} \cdot 2} = {(17 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(169 - x^{2})}^{\frac{1}{2} \cdot 2} = {(17 - x)}^{2}$

Rewrite the expression.

${(169 - x^{2})}^{1} = {(17 - x)}^{2}$

Simplify.

$169 - x^{2} = {(17 - x)}^{2}$

Simplify the right side.

Simplify ${(17 - x)}^{2}$ .

Rewrite ${(17 - x)}^{2}$ as $(17 - x) (17 - x)$ .

$169 - x^{2} = (17 - x) (17 - x)$

Expand $(17 - x) (17 - x)$ using the FOIL Method.

Apply the distributive property.

$169 - x^{2} = 17 (17 - x) - x (17 - x)$

Apply the distributive property.

$169 - x^{2} = 17 \cdot 17 + 17 (- x) - x (17 - x)$

Apply the distributive property.

$169 - x^{2} = 17 \cdot 17 + 17 (- x) - x \cdot 17 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $17$ by $17$ .

$169 - x^{2} = 289 + 17 (- x) - x \cdot 17 - x (- x)$

Multiply $- 1$ by $17$ .

$169 - x^{2} = 289 - 17 x - x \cdot 17 - x (- x)$

Multiply $17$ by $- 1$ .

$169 - x^{2} = 289 - 17 x - 17 x - x (- x)$

Rewrite using the commutative property of multiplication.

$169 - x^{2} = 289 - 17 x - 17 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$169 - x^{2} = 289 - 17 x - 17 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$169 - x^{2} = 289 - 17 x - 17 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$169 - x^{2} = 289 - 17 x - 17 x + 1 x^{2}$

Multiply $x^{2}$ by $1$ .

$169 - x^{2} = 289 - 17 x - 17 x + x^{2}$

Subtract $17 x$ from $- 17 x$ .

$169 - x^{2} = 289 - 34 x + x^{2}$

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.

$289 - 34 x + x^{2} = 169 - x^{2}$

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

Add $x^{2}$ to both sides of the equation.

$289 - 34 x + x^{2} + x^{2} = 169$

Add $x^{2}$ and $x^{2}$ .

$289 - 34 x + 2 x^{2} = 169$

Subtract $169$ from both sides of the equation.

$289 - 34 x + 2 x^{2} - 169 = 0$

Subtract $169$ from $289$ .

$- 34 x + 2 x^{2} + 120 = 0$

Factor the left side of the equation.

Factor $2$ out of $- 34 x + 2 x^{2} + 120$ .

Factor $2$ out of $- 34 x$ .

$2 (- 17 x) + 2 x^{2} + 120 = 0$

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

$2 (- 17 x) + 2 (x^{2}) + 120 = 0$

Factor $2$ out of $120$ .

$2 (- 17 x) + 2 x^{2} + 2 \cdot 60 = 0$

Factor $2$ out of $2 (- 17 x) + 2 x^{2}$ .

$2 (- 17 x + x^{2}) + 2 \cdot 60 = 0$

Factor $2$ out of $2 (- 17 x + x^{2}) + 2 \cdot 60$ .

$2 (- 17 x + x^{2} + 60) = 0$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2 (- 17 u + u^{2} + 60) = 0$

Factor $- 17 u + u^{2} + 60$ 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 $60$ and whose sum is $- 17$ .

$- 12, - 5$

Write the factored form using these integers.

$2 ((u - 12) (u - 5)) = 0$

Factor.

Replace all occurrences of $u$ with $x$ .

$2 ((x - 12) (x - 5)) = 0$

Remove unnecessary parentheses.

$2 (x - 12) (x - 5) = 0$

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

$x - 12 = 0$

$x - 5 = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $2 (x - 12) (x - 5) = 0$ true.

$x = 12, 5$

Exclude the solutions that do not make $x - \sqrt{169 - x^{2}} = 17$ true.

$No solution$

Solve for x x- square root of 169-x^2=17

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