Finite Math Examples

$4 (x^{2} - 16 x + 64) + 9 y^{2} = 36$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$4 x^{2} + 4 (- 16 x) + 4 \cdot 64 + 9 y^{2} = 36$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $- 16$ by $4$ .

$4 x^{2} - 64 x + 4 \cdot 64 + 9 y^{2} = 36$

Step 1.2.2

Multiply $4$ by $64$ .

$4 x^{2} - 64 x + 256 + 9 y^{2} = 36$

Step 2

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

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Step 2.1

Subtract $36$ from both sides of the equation.

$4 x^{2} - 64 x + 256 + 9 y^{2} - 36 = 0$

Step 2.2

Subtract $36$ from $256$ .

$4 x^{2} - 64 x + 9 y^{2} + 220 = 0$

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 $a = 4$ , $b = - 64$ , and $c = 9 y^{2} + 220$ into the quadratic formula and solve for $x$ .

$\frac{64 \pm \sqrt{{(- 64)}^{2} - 4 \cdot (4 \cdot (9 y^{2} + 220))}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 64$ to the power of $2$ .

$x = \frac{64 \pm \sqrt{4096 - 4 \cdot 4 \cdot (9 y^{2} + 220)}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4$ by $4$ .

$x = \frac{64 \pm \sqrt{4096 - 16 \cdot (9 y^{2} + 220)}}{2 \cdot 4}$

Step 5.1.3

Apply the distributive property.

$x = \frac{64 \pm \sqrt{4096 - 16 (9 y^{2}) - 16 \cdot 220}}{2 \cdot 4}$

Step 5.1.4

Multiply $9$ by $- 16$ .

$x = \frac{64 \pm \sqrt{4096 - 144 y^{2} - 16 \cdot 220}}{2 \cdot 4}$

Step 5.1.5

Multiply $- 16$ by $220$ .

$x = \frac{64 \pm \sqrt{4096 - 144 y^{2} - 3520}}{2 \cdot 4}$

Step 5.1.6

Subtract $3520$ from $4096$ .

$x = \frac{64 \pm \sqrt{- 144 y^{2} + 576}}{2 \cdot 4}$

Step 5.1.7

Rewrite $- 144 y^{2} + 576$ in a factored form.

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Step 5.1.7.1

Factor $144$ out of $- 144 y^{2} + 576$ .

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Step 5.1.7.1.1

Factor $144$ out of $- 144 y^{2}$ .

$x = \frac{64 \pm \sqrt{144 (- y^{2}) + 576}}{2 \cdot 4}$

Step 5.1.7.1.2

Factor $144$ out of $576$ .

$x = \frac{64 \pm \sqrt{144 (- y^{2}) + 144 (4)}}{2 \cdot 4}$

Step 5.1.7.1.3

Factor $144$ out of $144 (- y^{2}) + 144 (4)$ .

$x = \frac{64 \pm \sqrt{144 (- y^{2} + 4)}}{2 \cdot 4}$

Step 5.1.7.2

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

$x = \frac{64 \pm \sqrt{144 (- y^{2} + 2^{2})}}{2 \cdot 4}$

Step 5.1.7.3

Reorder $- y^{2}$ and $2^{2}$ .

$x = \frac{64 \pm \sqrt{144 (2^{2} - y^{2})}}{2 \cdot 4}$

Step 5.1.7.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = y$ .

$x = \frac{64 \pm \sqrt{144 (2 + y) (2 - y)}}{2 \cdot 4}$

Step 5.1.8

Rewrite $144 (2 + y) (2 - y)$ as $12^{2} ((2 + y) (2 - y))$ .

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Step 5.1.8.1

Rewrite $144$ as $12^{2}$ .

$x = \frac{64 \pm \sqrt{12^{2} (2 + y) (2 - y)}}{2 \cdot 4}$

Step 5.1.8.2

Add parentheses.

$x = \frac{64 \pm \sqrt{12^{2} ((2 + y) (2 - y))}}{2 \cdot 4}$

Step 5.1.9

Pull terms out from under the radical.

$x = \frac{64 \pm 12 \sqrt{(2 + y) (2 - y)}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{64 \pm 12 \sqrt{(2 + y) (2 - y)}}{8}$

Step 5.3

Simplify $\frac{64 \pm 12 \sqrt{(2 + y) (2 - y)}}{8}$ .

$x = \frac{16 \pm 3 \sqrt{(2 + y) (2 - y)}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{16 + 3 \sqrt{(2 + y) (2 - y)}}{2}$

$x = \frac{16 - 3 \sqrt{(2 + y) (2 - y)}}{2}$