Find the Points of Intersection y = square root of 196-x^2 , y=16-x

$y = \sqrt{196 - x^{2}}$ , $y = 16 - x$

Step 1

Eliminate the equal sides of each equation and combine.

$\sqrt{196 - x^{2}} = 16 - x$

Step 2

Solve $\sqrt{196 - x^{2}} = 16 - x$ for $x$ .

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

${\sqrt{196 - x^{2}}}^{2} = {(16 - x)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{196 - x^{2}}$ as ${(196 - x^{2})}^{\frac{1}{2}}$ .

${({(196 - x^{2})}^{\frac{1}{2}})}^{2} = {(16 - x)}^{2}$

Simplify the left side.

Simplify ${({(196 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(196 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(196 - x^{2})}^{\frac{1}{2} \cdot 2} = {(16 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(196 - x^{2})}^{\frac{1}{2} \cdot 2} = {(16 - x)}^{2}$

Rewrite the expression.

${(196 - x^{2})}^{1} = {(16 - x)}^{2}$

Simplify.

$196 - x^{2} = {(16 - x)}^{2}$

Simplify the right side.

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

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

$196 - x^{2} = (16 - x) (16 - x)$

Expand $(16 - x) (16 - x)$ using the FOIL Method.

Apply the distributive property.

$196 - x^{2} = 16 (16 - x) - x (16 - x)$

Apply the distributive property.

$196 - x^{2} = 16 \cdot 16 + 16 (- x) - x (16 - x)$

Apply the distributive property.

$196 - x^{2} = 16 \cdot 16 + 16 (- x) - x \cdot 16 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $16$ by $16$ .

$196 - x^{2} = 256 + 16 (- x) - x \cdot 16 - x (- x)$

Multiply $- 1$ by $16$ .

$196 - x^{2} = 256 - 16 x - x \cdot 16 - x (- x)$

Multiply $16$ by $- 1$ .

$196 - x^{2} = 256 - 16 x - 16 x - x (- x)$

Rewrite using the commutative property of multiplication.

$196 - x^{2} = 256 - 16 x - 16 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$196 - x^{2} = 256 - 16 x - 16 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$196 - x^{2} = 256 - 16 x - 16 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$196 - x^{2} = 256 - 16 x - 16 x + 1 x^{2}$

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

$196 - x^{2} = 256 - 16 x - 16 x + x^{2}$

Subtract $16 x$ from $- 16 x$ .

$196 - x^{2} = 256 - 32 x + x^{2}$

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.

$256 - 32 x + x^{2} = 196 - x^{2}$

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

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

$256 - 32 x + x^{2} + x^{2} = 196$

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

$256 - 32 x + 2 x^{2} = 196$

Subtract $196$ from both sides of the equation.

$256 - 32 x + 2 x^{2} - 196 = 0$

Subtract $196$ from $256$ .

$- 32 x + 2 x^{2} + 60 = 0$

Factor the left side of the equation.

Factor $2$ out of $- 32 x + 2 x^{2} + 60$ .

Factor $2$ out of $- 32 x$ .

$2 (- 16 x) + 2 x^{2} + 60 = 0$

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

$2 (- 16 x) + 2 (x^{2}) + 60 = 0$

Factor $2$ out of $60$ .

$2 (- 16 x) + 2 x^{2} + 2 \cdot 30 = 0$

Factor $2$ out of $2 (- 16 x) + 2 x^{2}$ .

$2 (- 16 x + x^{2}) + 2 \cdot 30 = 0$

Factor $2$ out of $2 (- 16 x + x^{2}) + 2 \cdot 30$ .

$2 (- 16 x + x^{2} + 30) = 0$

Reorder terms.

$2 (x^{2} - 16 x + 30) = 0$

Divide each term in $2 (x^{2} - 16 x + 30) = 0$ by $2$ and simplify.

Divide each term in $2 (x^{2} - 16 x + 30) = 0$ by $2$ .

$\frac{2 (x^{2} - 16 x + 30)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (x^{2} - 16 x + 30)}{2} = \frac{0}{2}$

Divide $x^{2} - 16 x + 30$ by $1$ .

$x^{2} - 16 x + 30 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x^{2} - 16 x + 30 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 16 - x$

Substitute the values $a = 1$ , $b = - 16$ , and $c = 30$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 30)}}{2 \cdot 1} + y = 16 - x$

Simplify.

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 30$ .

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 30}}{2 \cdot 1}$

Multiply $- 4$ by $30$ .

$x = \frac{16 \pm \sqrt{256 - 120}}{2 \cdot 1}$

Subtract $120$ from $256$ .

$x = \frac{16 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{16 \pm \sqrt{4 (34)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{16 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{16 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{16 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{34}}{2}$ .

$x = 8 \pm \sqrt{34}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 30$ .

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 30}}{2 \cdot 1}$

Multiply $- 4$ by $30$ .

$x = \frac{16 \pm \sqrt{256 - 120}}{2 \cdot 1}$

Subtract $120$ from $256$ .

$x = \frac{16 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{16 \pm \sqrt{4 (34)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{16 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{16 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{16 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{34}}{2}$ .

$x = 8 \pm \sqrt{34}$

Change the $\pm$ to $+$ .

$x = 8 + \sqrt{34}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 30$ .

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 30}}{2 \cdot 1}$

Multiply $- 4$ by $30$ .

$x = \frac{16 \pm \sqrt{256 - 120}}{2 \cdot 1}$

Subtract $120$ from $256$ .

$x = \frac{16 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{16 \pm \sqrt{4 (34)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{16 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{16 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{16 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{34}}{2}$ .

$x = 8 \pm \sqrt{34}$

Change the $\pm$ to $-$ .

$x = 8 - \sqrt{34}$

The final answer is the combination of both solutions.

$x = 8 + \sqrt{34}, 8 - \sqrt{34}$

Step 3

Evaluate $y$ when $x = 8 + \sqrt{34}$ .

Substitute $8 + \sqrt{34}$ for $x$ .

$y = 16 - (8 + \sqrt{34})$

Simplify $16 - (8 + \sqrt{34})$ .

Simplify each term.

Apply the distributive property.

$y = 16 - 1 \cdot 8 - \sqrt{34}$

Multiply $- 1$ by $8$ .

$y = 16 - 8 - \sqrt{34}$

Subtract $8$ from $16$ .

$y = 8 - \sqrt{34}$

Step 4

Evaluate $y$ when $x = 8 - \sqrt{34}$ .

Substitute $8 - \sqrt{34}$ for $x$ .

$y = 16 - (8 - \sqrt{34})$

Simplify $16 - (8 - \sqrt{34})$ .

Simplify each term.

Apply the distributive property.

$y = 16 - 1 \cdot 8 - - \sqrt{34}$

Multiply $- 1$ by $8$ .

$y = 16 - 8 - - \sqrt{34}$

Multiply $- - \sqrt{34}$ .

Multiply $- 1$ by $- 1$ .

$y = 16 - 8 + 1 \sqrt{34}$

Multiply $\sqrt{34}$ by $1$ .

$y = 16 - 8 + \sqrt{34}$

Subtract $8$ from $16$ .

$y = 8 + \sqrt{34}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(8 + \sqrt{34}, 8 - \sqrt{34})$

$(8 - \sqrt{34}, 8 + \sqrt{34})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(8 + \sqrt{34}, 8 - \sqrt{34}), (8 - \sqrt{34}, 8 + \sqrt{34})$

Equation Form:

$x = 8 + \sqrt{34}, y = 8 - \sqrt{34}$

$x = 8 - \sqrt{34}, y = 8 + \sqrt{34}$

Step 7

Find the Points of Intersection y = square root of 196-x^2 , y=16-x

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