Finite Math Examples

$r^{2} = {(4 - h)}^{2} + {(2 - k)}^{2}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract ${(4 - h)}^{2}$ from both sides of the equation.

$r^{2} - {(4 - h)}^{2} = {(2 - k)}^{2}$

Step 1.2

Subtract ${(2 - k)}^{2}$ from both sides of the equation.

$r^{2} - {(4 - h)}^{2} - {(2 - k)}^{2} = 0$

Step 2

Simplify $r^{2} - {(4 - h)}^{2} - {(2 - k)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(4 - h)}^{2}$ as $(4 - h) (4 - h)$ .

$r^{2} - ((4 - h) (4 - h)) - {(2 - k)}^{2} = 0$

Step 2.1.2

Expand $(4 - h) (4 - h)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$r^{2} - (4 (4 - h) - h (4 - h)) - {(2 - k)}^{2} = 0$

Step 2.1.2.2

Apply the distributive property.

$r^{2} - (4 \cdot 4 + 4 (- h) - h (4 - h)) - {(2 - k)}^{2} = 0$

Step 2.1.2.3

Apply the distributive property.

$r^{2} - (4 \cdot 4 + 4 (- h) - h \cdot 4 - h (- h)) - {(2 - k)}^{2} = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $4$ by $4$ .

$r^{2} - (16 + 4 (- h) - h \cdot 4 - h (- h)) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.2

Multiply $- 1$ by $4$ .

$r^{2} - (16 - 4 h - h \cdot 4 - h (- h)) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.3

Multiply $4$ by $- 1$ .

$r^{2} - (16 - 4 h - 4 h - h (- h)) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.4

Rewrite using the commutative property of multiplication.

$r^{2} - (16 - 4 h - 4 h - 1 \cdot - 1 h \cdot h) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.5

Multiply $h$ by $h$ by adding the exponents.

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Step 2.1.3.1.5.1

Move $h$ .

$r^{2} - (16 - 4 h - 4 h - 1 \cdot - 1 (h \cdot h)) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.5.2

Multiply $h$ by $h$ .

$r^{2} - (16 - 4 h - 4 h - 1 \cdot - 1 h^{2}) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.6

Multiply $- 1$ by $- 1$ .

$r^{2} - (16 - 4 h - 4 h + 1 h^{2}) - {(2 - k)}^{2} = 0$

Step 2.1.3.1.7

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

$r^{2} - (16 - 4 h - 4 h + h^{2}) - {(2 - k)}^{2} = 0$

Step 2.1.3.2

Subtract $4 h$ from $- 4 h$ .

$r^{2} - (16 - 8 h + h^{2}) - {(2 - k)}^{2} = 0$

Step 2.1.4

Apply the distributive property.

$r^{2} - 1 \cdot 16 - (- 8 h) - h^{2} - {(2 - k)}^{2} = 0$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $- 1$ by $16$ .

$r^{2} - 16 - (- 8 h) - h^{2} - {(2 - k)}^{2} = 0$

Step 2.1.5.2

Multiply $- 8$ by $- 1$ .

$r^{2} - 16 + 8 h - h^{2} - {(2 - k)}^{2} = 0$

Step 2.1.6

Rewrite ${(2 - k)}^{2}$ as $(2 - k) (2 - k)$ .

$r^{2} - 16 + 8 h - h^{2} - ((2 - k) (2 - k)) = 0$

Step 2.1.7

Expand $(2 - k) (2 - k)$ using the FOIL Method.

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Step 2.1.7.1

Apply the distributive property.

$r^{2} - 16 + 8 h - h^{2} - (2 (2 - k) - k (2 - k)) = 0$

Step 2.1.7.2

Apply the distributive property.

$r^{2} - 16 + 8 h - h^{2} - (2 \cdot 2 + 2 (- k) - k (2 - k)) = 0$

Step 2.1.7.3

Apply the distributive property.

$r^{2} - 16 + 8 h - h^{2} - (2 \cdot 2 + 2 (- k) - k \cdot 2 - k (- k)) = 0$

Step 2.1.8

Simplify and combine like terms.

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Step 2.1.8.1

Simplify each term.

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Step 2.1.8.1.1

Multiply $2$ by $2$ .

$r^{2} - 16 + 8 h - h^{2} - (4 + 2 (- k) - k \cdot 2 - k (- k)) = 0$

Step 2.1.8.1.2

Multiply $- 1$ by $2$ .

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - k \cdot 2 - k (- k)) = 0$

Step 2.1.8.1.3

Multiply $2$ by $- 1$ .

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - 2 k - k (- k)) = 0$

Step 2.1.8.1.4

Rewrite using the commutative property of multiplication.

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - 2 k - 1 \cdot - 1 k \cdot k) = 0$

Step 2.1.8.1.5

Multiply $k$ by $k$ by adding the exponents.

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Step 2.1.8.1.5.1

Move $k$ .

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - 2 k - 1 \cdot - 1 (k \cdot k)) = 0$

Step 2.1.8.1.5.2

Multiply $k$ by $k$ .

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - 2 k - 1 \cdot - 1 k^{2}) = 0$

Step 2.1.8.1.6

Multiply $- 1$ by $- 1$ .

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - 2 k + 1 k^{2}) = 0$

Step 2.1.8.1.7

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

$r^{2} - 16 + 8 h - h^{2} - (4 - 2 k - 2 k + k^{2}) = 0$

Step 2.1.8.2

Subtract $2 k$ from $- 2 k$ .

$r^{2} - 16 + 8 h - h^{2} - (4 - 4 k + k^{2}) = 0$

Step 2.1.9

Apply the distributive property.

$r^{2} - 16 + 8 h - h^{2} - 1 \cdot 4 - (- 4 k) - k^{2} = 0$

Step 2.1.10

Simplify.

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Step 2.1.10.1

Multiply $- 1$ by $4$ .

$r^{2} - 16 + 8 h - h^{2} - 4 - (- 4 k) - k^{2} = 0$

Step 2.1.10.2

Multiply $- 4$ by $- 1$ .

$r^{2} - 16 + 8 h - h^{2} - 4 + 4 k - k^{2} = 0$

Step 2.2

Subtract $4$ from $- 16$ .

$r^{2} + 8 h - h^{2} - 20 + 4 k - k^{2} = 0$

Step 3

Move all terms not containing $r$ to the right side of the equation.

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Step 3.1

Subtract $8 h$ from both sides of the equation.

$r^{2} - h^{2} - 20 + 4 k - k^{2} = - 8 h$

Step 3.2

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

$r^{2} - 20 + 4 k - k^{2} = - 8 h + h^{2}$

Step 3.3

Add $20$ to both sides of the equation.

$r^{2} + 4 k - k^{2} = - 8 h + h^{2} + 20$

Step 3.4

Subtract $4 k$ from both sides of the equation.

$r^{2} - k^{2} = - 8 h + h^{2} + 20 - 4 k$

Step 3.5

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

$r^{2} = - 8 h + h^{2} + 20 - 4 k + k^{2}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$r = \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$

$r = - \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$

$r = \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$

$r = - \sqrt{- 8 h + h^{2} + 20 - 4 k + k^{2}}$