Finite Math Examples

{(x - 3)}^{2} + {(y - 5)}^{2} = r^{2}

${(x - 3)}^{2} + {(y - 5)}^{2} = r^{2}$

Step 1

Subtract

r^{2}

$r^{2}$ from both sides of the equation.

{(x - 3)}^{2} + {(y - 5)}^{2} - r^{2} = 0

${(x - 3)}^{2} + {(y - 5)}^{2} - r^{2} = 0$

Step 2

Simplify

{(x - 3)}^{2} + {(y - 5)}^{2} - r^{2}

${(x - 3)}^{2} + {(y - 5)}^{2} - r^{2}$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite

{(x - 3)}^{2}

${(x - 3)}^{2}$ as

(x - 3) (x - 3)

$(x - 3) (x - 3)$ .

(x - 3) (x - 3) + {(y - 5)}^{2} - r^{2} = 0

$(x - 3) (x - 3) + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.2

Expand

(x - 3) (x - 3)

$(x - 3) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 2.1.2.1

Apply the distributive property.

x (x - 3) - 3 (x - 3) + {(y - 5)}^{2} - r^{2} = 0

$x (x - 3) - 3 (x - 3) + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.2.2

Apply the distributive property.

x \cdot x + x \cdot - 3 - 3 (x - 3) + {(y - 5)}^{2} - r^{2} = 0

$x \cdot x + x \cdot - 3 - 3 (x - 3) + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.2.3

Apply the distributive property.

x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0$

x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.3.1.2

Move

- 3

$- 3$ to the left of

x

$x$ .

x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.3.1.3

Multiply

- 3

$- 3$ by

- 3

$- 3$ .

x^{2} - 3 x - 3 x + 9 + {(y - 5)}^{2} - r^{2} = 0

$x^{2} - 3 x - 3 x + 9 + {(y - 5)}^{2} - r^{2} = 0$

x^{2} - 3 x - 3 x + 9 + {(y - 5)}^{2} - r^{2} = 0

$x^{2} - 3 x - 3 x + 9 + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.3.2

Subtract

3 x

$3 x$ from

- 3 x

$- 3 x$ .

x^{2} - 6 x + 9 + {(y - 5)}^{2} - r^{2} = 0

$x^{2} - 6 x + 9 + {(y - 5)}^{2} - r^{2} = 0$

x^{2} - 6 x + 9 + {(y - 5)}^{2} - r^{2} = 0

$x^{2} - 6 x + 9 + {(y - 5)}^{2} - r^{2} = 0$

Step 2.1.4

Rewrite

{(y - 5)}^{2}

${(y - 5)}^{2}$ as

(y - 5) (y - 5)

$(y - 5) (y - 5)$ .

x^{2} - 6 x + 9 + (y - 5) (y - 5) - r^{2} = 0

$x^{2} - 6 x + 9 + (y - 5) (y - 5) - r^{2} = 0$

Step 2.1.5

Expand

(y - 5) (y - 5)

$(y - 5) (y - 5)$ using the FOIL Method.

Tap for more steps...

Step 2.1.5.1

Apply the distributive property.

x^{2} - 6 x + 9 + y (y - 5) - 5 (y - 5) - r^{2} = 0

$x^{2} - 6 x + 9 + y (y - 5) - 5 (y - 5) - r^{2} = 0$

Step 2.1.5.2

Apply the distributive property.

x^{2} - 6 x + 9 + y \cdot y + y \cdot - 5 - 5 (y - 5) - r^{2} = 0

$x^{2} - 6 x + 9 + y \cdot y + y \cdot - 5 - 5 (y - 5) - r^{2} = 0$

Step 2.1.5.3

Apply the distributive property.

x^{2} - 6 x + 9 + y \cdot y + y \cdot - 5 - 5 y - 5 \cdot - 5 - r^{2} = 0

$x^{2} - 6 x + 9 + y \cdot y + y \cdot - 5 - 5 y - 5 \cdot - 5 - r^{2} = 0$

x^{2} - 6 x + 9 + y \cdot y + y \cdot - 5 - 5 y - 5 \cdot - 5 - r^{2} = 0

$x^{2} - 6 x + 9 + y \cdot y + y \cdot - 5 - 5 y - 5 \cdot - 5 - r^{2} = 0$

Step 2.1.6

Simplify and combine like terms.

Tap for more steps...

Step 2.1.6.1

Simplify each term.

Tap for more steps...

Step 2.1.6.1.1

Multiply

y

$y$ by

y

$y$ .

x^{2} - 6 x + 9 + y^{2} + y \cdot - 5 - 5 y - 5 \cdot - 5 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} + y \cdot - 5 - 5 y - 5 \cdot - 5 - r^{2} = 0$

Step 2.1.6.1.2

Move

- 5

$- 5$ to the left of

y

$y$ .

x^{2} - 6 x + 9 + y^{2} - 5 \cdot y - 5 y - 5 \cdot - 5 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} - 5 \cdot y - 5 y - 5 \cdot - 5 - r^{2} = 0$

Step 2.1.6.1.3

Multiply

- 5

$- 5$ by

- 5

$- 5$ .

x^{2} - 6 x + 9 + y^{2} - 5 y - 5 y + 25 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} - 5 y - 5 y + 25 - r^{2} = 0$

x^{2} - 6 x + 9 + y^{2} - 5 y - 5 y + 25 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} - 5 y - 5 y + 25 - r^{2} = 0$

Step 2.1.6.2

Subtract

5 y

$5 y$ from

- 5 y

$- 5 y$ .

x^{2} - 6 x + 9 + y^{2} - 10 y + 25 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} - 10 y + 25 - r^{2} = 0$

x^{2} - 6 x + 9 + y^{2} - 10 y + 25 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} - 10 y + 25 - r^{2} = 0$

x^{2} - 6 x + 9 + y^{2} - 10 y + 25 - r^{2} = 0

$x^{2} - 6 x + 9 + y^{2} - 10 y + 25 - r^{2} = 0$

Step 2.2

Add

9

$9$ and

25

$25$ .

x^{2} - 6 x + y^{2} - 10 y + 34 - r^{2} = 0

$x^{2} - 6 x + y^{2} - 10 y + 34 - r^{2} = 0$

x^{2} - 6 x + y^{2} - 10 y + 34 - r^{2} = 0

$x^{2} - 6 x + y^{2} - 10 y + 34 - r^{2} = 0$

Step 3

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values

a = 1

$a = 1$ ,

b = - 6

$b = - 6$ , and

c = y^{2} - 10 y + 34 - r^{2}

$c = y^{2} - 10 y + 34 - r^{2}$ into the quadratic formula and solve for

x

$x$ .

\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot (y^{2} - 10 y + 34 - r^{2}))}}{2 \cdot 1}

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot (y^{2} - 10 y + 34 - r^{2}))}}{2 \cdot 1}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise

- 6

$- 6$ to the power of

2

$2$ .

x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}$

Step 5.1.2

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{6 \pm \sqrt{36 - 4 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}

$x = \frac{6 \pm \sqrt{36 - 4 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 4 (- 10 y) - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 4 (- 10 y) - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}$

Step 5.1.4

Simplify.

Tap for more steps...

Step 5.1.4.1

Multiply

- 10

$- 10$ by

- 4

$- 4$ .

x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}$

Step 5.1.4.2

Multiply

$- 4$ by

$34$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 136 - 4 (- r^{2})}}{2 \cdot 1}$

Step 5.1.4.3

Multiply

$- 1$ by

$- 4$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 136 + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.5

Subtract

$136$ from

$36$ .

$x = \frac{6 \pm \sqrt{- 4 y^{2} + 40 y - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.6

Rewrite

$- 4 y^{2} + 40 y - 100 + 4 r^{2}$ in a factored form.

Tap for more steps...

Step 5.1.6.1

Factor

$4$ out of

$- 4 y^{2} + 40 y - 100 + 4 r^{2}$ .

Tap for more steps...

Step 5.1.6.1.1

Factor

$4$ out of

$- 4 y^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 40 y - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.6.1.2

Factor

$4$ out of

$40 y$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (10 y) - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.6.1.3

Factor

$4$ out of

$- 100$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (10 y) + 4 \cdot - 25 + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.6.1.4

Factor

$4$ out of

$4 (- y^{2}) + 4 (10 y)$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y) + 4 \cdot - 25 + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.6.1.5

Factor

$4$ out of

$4 (- y^{2} + 10 y) + 4 \cdot - 25$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y - 25) + 4 r^{2}}}{2 \cdot 1}$

Step 5.1.6.1.6

Factor

$4$ out of

$4 (- y^{2} + 10 y - 25) + 4 r^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y - 25 + r^{2})}}{2 \cdot 1}$

Step 5.1.6.2

Rewrite

$y^{2} - 10 y + 25$ as

${(y - 5)}^{2}$ .

Tap for more steps...

Step 5.1.6.2.1

Rewrite

$25$ as

$5^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- (y^{2} - 10 y + 5^{2}) + r^{2})}}{2 \cdot 1}$

Step 5.1.6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 y = 2 \cdot y \cdot 5$

Step 5.1.6.2.3

Rewrite the polynomial.

$x = \frac{6 \pm \sqrt{4 (- (y^{2} - 2 \cdot y \cdot 5 + 5^{2}) + r^{2})}}{2 \cdot 1}$

Step 5.1.6.2.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = y$ and

$b = 5$ .

$x = \frac{6 \pm \sqrt{4 (- {(y - 5)}^{2} + r^{2})}}{2 \cdot 1}$

Step 5.1.6.3

Reorder

$- {(y - 5)}^{2}$ and

$r^{2}$ .

$x = \frac{6 \pm \sqrt{4 (r^{2} - {(y - 5)}^{2})}}{2 \cdot 1}$

Step 5.1.6.4

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = r$ and

$b = y - 5$ .

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - (y - 5)))}}{2 \cdot 1}$

Step 5.1.6.5

Simplify.

Tap for more steps...

Step 5.1.6.5.1

Apply the distributive property.

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

Step 5.1.6.5.2

Multiply

$- 1$ by

$- 5$ .

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

$x = \frac{6 \pm \sqrt{4 (r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 5.1.7

Rewrite

$4 (r + y - 5) (r - y + 5)$ as

$2^{2} ((r + y - 5) (r - y + 5))$ .

Tap for more steps...

Step 5.1.7.1

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} (r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 5.1.7.2

Add parentheses.

$x = \frac{6 \pm \sqrt{2^{2} ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 5.2

Multiply

$2$ by

$1$ .

$x = \frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2}$

Step 5.3

Simplify

$\frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2}$ .

$x = 3 \pm \sqrt{(r + y - 5) (r - y + 5)}$

Step 6

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Raise

$- 6$ to the power of

$2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}$

Step 6.1.2

Multiply

$- 4$ by

$1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}$

Step 6.1.3

Apply the distributive property.

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 4 (- 10 y) - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}$

Step 6.1.4

Simplify.

Tap for more steps...

Step 6.1.4.1

Multiply

$- 10$ by

$- 4$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}$

Step 6.1.4.2

Multiply

$- 4$ by

$34$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 136 - 4 (- r^{2})}}{2 \cdot 1}$

Step 6.1.4.3

Multiply

$- 1$ by

$- 4$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 136 + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.5

Subtract

$136$ from

$36$ .

$x = \frac{6 \pm \sqrt{- 4 y^{2} + 40 y - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.6

Rewrite

$- 4 y^{2} + 40 y - 100 + 4 r^{2}$ in a factored form.

Tap for more steps...

Step 6.1.6.1

Factor

$4$ out of

$- 4 y^{2} + 40 y - 100 + 4 r^{2}$ .

Tap for more steps...

Step 6.1.6.1.1

Factor

$4$ out of

$- 4 y^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 40 y - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.6.1.2

Factor

$4$ out of

$40 y$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (10 y) - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.6.1.3

Factor

$4$ out of

$- 100$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (10 y) + 4 \cdot - 25 + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.6.1.4

Factor

$4$ out of

$4 (- y^{2}) + 4 (10 y)$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y) + 4 \cdot - 25 + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.6.1.5

Factor

$4$ out of

$4 (- y^{2} + 10 y) + 4 \cdot - 25$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y - 25) + 4 r^{2}}}{2 \cdot 1}$

Step 6.1.6.1.6

Factor

$4$ out of

$4 (- y^{2} + 10 y - 25) + 4 r^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y - 25 + r^{2})}}{2 \cdot 1}$

Step 6.1.6.2

Rewrite

$y^{2} - 10 y + 25$ as

${(y - 5)}^{2}$ .

Tap for more steps...

Step 6.1.6.2.1

Rewrite

$25$ as

$5^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- (y^{2} - 10 y + 5^{2}) + r^{2})}}{2 \cdot 1}$

Step 6.1.6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 y = 2 \cdot y \cdot 5$

Step 6.1.6.2.3

Rewrite the polynomial.

$x = \frac{6 \pm \sqrt{4 (- (y^{2} - 2 \cdot y \cdot 5 + 5^{2}) + r^{2})}}{2 \cdot 1}$

Step 6.1.6.2.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = y$ and

$b = 5$ .

$x = \frac{6 \pm \sqrt{4 (- {(y - 5)}^{2} + r^{2})}}{2 \cdot 1}$

Step 6.1.6.3

Reorder

$- {(y - 5)}^{2}$ and

$r^{2}$ .

$x = \frac{6 \pm \sqrt{4 (r^{2} - {(y - 5)}^{2})}}{2 \cdot 1}$

Step 6.1.6.4

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = r$ and

$b = y - 5$ .

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - (y - 5)))}}{2 \cdot 1}$

Step 6.1.6.5

Simplify.

Tap for more steps...

Step 6.1.6.5.1

Apply the distributive property.

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

Step 6.1.6.5.2

Multiply

$- 1$ by

$- 5$ .

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

$x = \frac{6 \pm \sqrt{4 (r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 6.1.7

Rewrite

$4 (r + y - 5) (r - y + 5)$ as

$2^{2} ((r + y - 5) (r - y + 5))$ .

Tap for more steps...

Step 6.1.7.1

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} (r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 6.1.7.2

Add parentheses.

$x = \frac{6 \pm \sqrt{2^{2} ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 6.2

Multiply

$2$ by

$1$ .

$x = \frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2}$

Step 6.3

Simplify

$\frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2}$ .

$x = 3 \pm \sqrt{(r + y - 5) (r - y + 5)}$

Step 6.4

Change the

$\pm$ to

$+$ .

$x = 3 + \sqrt{(r + y - 5) (r - y + 5)}$

Step 7

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Raise

$- 6$ to the power of

$2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}$

Step 7.1.2

Multiply

$- 4$ by

$1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot (y^{2} - 10 y + 34 - r^{2})}}{2 \cdot 1}$

Step 7.1.3

Apply the distributive property.

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 4 (- 10 y) - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}$

Step 7.1.4

Simplify.

Tap for more steps...

Step 7.1.4.1

Multiply

$- 10$ by

$- 4$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 4 \cdot 34 - 4 (- r^{2})}}{2 \cdot 1}$

Step 7.1.4.2

Multiply

$- 4$ by

$34$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 136 - 4 (- r^{2})}}{2 \cdot 1}$

Step 7.1.4.3

Multiply

$- 1$ by

$- 4$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} + 40 y - 136 + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.5

Subtract

$136$ from

$36$ .

$x = \frac{6 \pm \sqrt{- 4 y^{2} + 40 y - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.6

Rewrite

$- 4 y^{2} + 40 y - 100 + 4 r^{2}$ in a factored form.

Tap for more steps...

Step 7.1.6.1

Factor

$4$ out of

$- 4 y^{2} + 40 y - 100 + 4 r^{2}$ .

Tap for more steps...

Step 7.1.6.1.1

Factor

$4$ out of

$- 4 y^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 40 y - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.6.1.2

Factor

$4$ out of

$40 y$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (10 y) - 100 + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.6.1.3

Factor

$4$ out of

$- 100$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (10 y) + 4 \cdot - 25 + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.6.1.4

Factor

$4$ out of

$4 (- y^{2}) + 4 (10 y)$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y) + 4 \cdot - 25 + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.6.1.5

Factor

$4$ out of

$4 (- y^{2} + 10 y) + 4 \cdot - 25$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y - 25) + 4 r^{2}}}{2 \cdot 1}$

Step 7.1.6.1.6

Factor

$4$ out of

$4 (- y^{2} + 10 y - 25) + 4 r^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2} + 10 y - 25 + r^{2})}}{2 \cdot 1}$

Step 7.1.6.2

Rewrite

$y^{2} - 10 y + 25$ as

${(y - 5)}^{2}$ .

Tap for more steps...

Step 7.1.6.2.1

Rewrite

$25$ as

$5^{2}$ .

$x = \frac{6 \pm \sqrt{4 (- (y^{2} - 10 y + 5^{2}) + r^{2})}}{2 \cdot 1}$

Step 7.1.6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 y = 2 \cdot y \cdot 5$

Step 7.1.6.2.3

Rewrite the polynomial.

$x = \frac{6 \pm \sqrt{4 (- (y^{2} - 2 \cdot y \cdot 5 + 5^{2}) + r^{2})}}{2 \cdot 1}$

Step 7.1.6.2.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = y$ and

$b = 5$ .

$x = \frac{6 \pm \sqrt{4 (- {(y - 5)}^{2} + r^{2})}}{2 \cdot 1}$

Step 7.1.6.3

Reorder

$- {(y - 5)}^{2}$ and

$r^{2}$ .

$x = \frac{6 \pm \sqrt{4 (r^{2} - {(y - 5)}^{2})}}{2 \cdot 1}$

Step 7.1.6.4

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = r$ and

$b = y - 5$ .

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - (y - 5)))}}{2 \cdot 1}$

Step 7.1.6.5

Simplify.

Tap for more steps...

Step 7.1.6.5.1

Apply the distributive property.

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

Step 7.1.6.5.2

Multiply

$- 1$ by

$- 5$ .

$x = \frac{6 \pm \sqrt{4 ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

$x = \frac{6 \pm \sqrt{4 (r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 7.1.7

Rewrite

$4 (r + y - 5) (r - y + 5)$ as

$2^{2} ((r + y - 5) (r - y + 5))$ .

Tap for more steps...

Step 7.1.7.1

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} (r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 7.1.7.2

Add parentheses.

$x = \frac{6 \pm \sqrt{2^{2} ((r + y - 5) (r - y + 5))}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2 \cdot 1}$

Step 7.2

Multiply

$2$ by

$1$ .

$x = \frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2}$

Step 7.3

Simplify

$\frac{6 \pm 2 \sqrt{(r + y - 5) (r - y + 5)}}{2}$ .

$x = 3 \pm \sqrt{(r + y - 5) (r - y + 5)}$

Step 7.4

Change the

$\pm$ to

$-$ .

$x = 3 - \sqrt{(r + y - 5) (r - y + 5)}$

Step 8

The final answer is the combination of both solutions.

$x = 3 + \sqrt{(r + y - 5) (r - y + 5)}$

$x = 3 - \sqrt{(r + y - 5) (r - y + 5)}$