Trigonometry Examples

$(h^{2} - 16 h + 64) + (k^{2} - 12 k + 36) = 100$

Step 1

Add $64$ and $36$ .

$h^{2} - 16 h + k^{2} - 12 k + 100 = 100$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $100$ from both sides of the equation.

$h^{2} - 16 h + k^{2} - 12 k + 100 - 100 = 0$

Step 2.2

Combine the opposite terms in $h^{2} - 16 h + k^{2} - 12 k + 100 - 100$ .

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Step 2.2.1

Subtract $100$ from $100$ .

$h^{2} - 16 h + k^{2} - 12 k + 0 = 0$

Step 2.2.2

Add $h^{2} - 16 h + k^{2} - 12 k$ and $0$ .

$h^{2} - 16 h + k^{2} - 12 k = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 16$ , and $c = k^{2} - 12 k$ into the quadratic formula and solve for $h$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot (k^{2} - 12 k))}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$h = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot (k^{2} - 12 k)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$h = \frac{16 \pm \sqrt{256 - 4 \cdot (k^{2} - 12 k)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$h = \frac{16 \pm \sqrt{256 - 4 k^{2} - 4 (- 12 k)}}{2 \cdot 1}$

Step 5.1.4

Multiply $- 12$ by $- 4$ .

$h = \frac{16 \pm \sqrt{256 - 4 k^{2} + 48 k}}{2 \cdot 1}$

Step 5.1.5

Reorder terms.

$h = \frac{16 \pm \sqrt{- 4 k^{2} + 48 k + 256}}{2 \cdot 1}$

Step 5.1.6

Rewrite $- 4 k^{2} + 48 k + 256$ in a factored form.

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Step 5.1.6.1

Factor $4$ out of $- 4 k^{2} + 48 k + 256$ .

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Step 5.1.6.1.1

Factor $4$ out of $- 4 k^{2}$ .

$h = \frac{16 \pm \sqrt{4 (- k^{2}) + 48 k + 256}}{2 \cdot 1}$

Step 5.1.6.1.2

Factor $4$ out of $48 k$ .

$h = \frac{16 \pm \sqrt{4 (- k^{2}) + 4 (12 k) + 256}}{2 \cdot 1}$

Step 5.1.6.1.3

Factor $4$ out of $256$ .

$h = \frac{16 \pm \sqrt{4 (- k^{2}) + 4 (12 k) + 4 (64)}}{2 \cdot 1}$

Step 5.1.6.1.4

Factor $4$ out of $4 (- k^{2}) + 4 (12 k)$ .

$h = \frac{16 \pm \sqrt{4 (- k^{2} + 12 k) + 4 (64)}}{2 \cdot 1}$

Step 5.1.6.1.5

Factor $4$ out of $4 (- k^{2} + 12 k) + 4 (64)$ .

$h = \frac{16 \pm \sqrt{4 (- k^{2} + 12 k + 64)}}{2 \cdot 1}$

Step 5.1.6.2

Factor by grouping.

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Step 5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 64 = - 64$ and whose sum is $b = 12$ .

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Step 5.1.6.2.1.1

Factor $12$ out of $12 k$ .

$h = \frac{16 \pm \sqrt{4 (- k^{2} + 12 (k) + 64)}}{2 \cdot 1}$

Step 5.1.6.2.1.2

Rewrite $12$ as $- 4$ plus $16$

$h = \frac{16 \pm \sqrt{4 (- k^{2} + (- 4 + 16) k + 64)}}{2 \cdot 1}$

Step 5.1.6.2.1.3

Apply the distributive property.

$h = \frac{16 \pm \sqrt{4 (- k^{2} - 4 k + 16 k + 64)}}{2 \cdot 1}$

Step 5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 5.1.6.2.2.1

Group the first two terms and the last two terms.

$h = \frac{16 \pm \sqrt{4 ((- k^{2} - 4 k) + 16 k + 64)}}{2 \cdot 1}$

Step 5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$h = \frac{16 \pm \sqrt{4 (k (- k - 4) - 16 (- k - 4))}}{2 \cdot 1}$

Step 5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- k - 4$ .

$h = \frac{16 \pm \sqrt{4 ((- k - 4) (k - 16))}}{2 \cdot 1}$

$h = \frac{16 \pm \sqrt{4 (- k - 4) (k - 16)}}{2 \cdot 1}$

Step 5.1.7

Rewrite $4 (- k - 4) (k - 16)$ as $2^{2} ((- k - 4) (k - 16))$ .

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Step 5.1.7.1

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

$h = \frac{16 \pm \sqrt{2^{2} (- k - 4) (k - 16)}}{2 \cdot 1}$

Step 5.1.7.2

Add parentheses.

$h = \frac{16 \pm \sqrt{2^{2} ((- k - 4) (k - 16))}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$h = \frac{16 \pm 2 \sqrt{(- k - 4) (k - 16)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$h = \frac{16 \pm 2 \sqrt{(- k - 4) (k - 16)}}{2}$

Step 5.3

Simplify $\frac{16 \pm 2 \sqrt{(- k - 4) (k - 16)}}{2}$ .

$h = 8 \pm \sqrt{(- k - 4) (k - 16)}$

Step 6

The final answer is the combination of both solutions.

$h = 8 + \sqrt{(- k - 4) (k - 16)}$

$h = 8 - \sqrt{(- k - 4) (k - 16)}$