Finite Math Examples

$4 x^{2} + 9 y^{2} - 32 x - 36 y + 64 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 4$ , $b = - 32$ , and $c = 9 y^{2} - 36 y + 64$ into the quadratic formula and solve for $x$ .

$\frac{32 \pm \sqrt{{(- 32)}^{2} - 4 \cdot (4 \cdot (9 y^{2} - 36 y + 64))}}{2 \cdot 4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 32$ to the power of $2$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot 4 \cdot (9 y^{2} - 36 y + 64)}}{2 \cdot 4}$

Step 3.1.2

Multiply $- 4$ by $4$ .

$x = \frac{32 \pm \sqrt{1024 - 16 \cdot (9 y^{2} - 36 y + 64)}}{2 \cdot 4}$

Step 3.1.3

Apply the distributive property.

$x = \frac{32 \pm \sqrt{1024 - 16 (9 y^{2}) - 16 (- 36 y) - 16 \cdot 64}}{2 \cdot 4}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $9$ by $- 16$ .

$x = \frac{32 \pm \sqrt{1024 - 144 y^{2} - 16 (- 36 y) - 16 \cdot 64}}{2 \cdot 4}$

Step 3.1.4.2

Multiply $- 36$ by $- 16$ .

$x = \frac{32 \pm \sqrt{1024 - 144 y^{2} + 576 y - 16 \cdot 64}}{2 \cdot 4}$

Step 3.1.4.3

Multiply $- 16$ by $64$ .

$x = \frac{32 \pm \sqrt{1024 - 144 y^{2} + 576 y - 1024}}{2 \cdot 4}$

Step 3.1.5

Subtract $1024$ from $1024$ .

$x = \frac{32 \pm \sqrt{- 144 y^{2} + 576 y + 0}}{2 \cdot 4}$

Step 3.1.6

Add $- 144 y^{2} + 576 y$ and $0$ .

$x = \frac{32 \pm \sqrt{- 144 y^{2} + 576 y}}{2 \cdot 4}$

Step 3.1.7

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

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Step 3.1.7.1

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

$x = \frac{32 \pm \sqrt{144 y (- y) + 576 y}}{2 \cdot 4}$

Step 3.1.7.2

Factor $144 y$ out of $576 y$ .

$x = \frac{32 \pm \sqrt{144 y (- y) + 144 y (4)}}{2 \cdot 4}$

Step 3.1.7.3

Factor $144 y$ out of $144 y (- y) + 144 y (4)$ .

$x = \frac{32 \pm \sqrt{144 y (- y + 4)}}{2 \cdot 4}$

Step 3.1.8

Rewrite $144 y (- y + 4)$ as $12^{2} (y (- y + 2^{2}))$ .

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Step 3.1.8.1

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

$x = \frac{32 \pm \sqrt{12^{2} y (- y + 4)}}{2 \cdot 4}$

Step 3.1.8.2

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

$x = \frac{32 \pm \sqrt{12^{2} y (- y + 2^{2})}}{2 \cdot 4}$

Step 3.1.8.3

Add parentheses.

$x = \frac{32 \pm \sqrt{12^{2} (y (- y + 2^{2}))}}{2 \cdot 4}$

Step 3.1.9

Pull terms out from under the radical.

$x = \frac{32 \pm 12 \sqrt{y (- y + 2^{2})}}{2 \cdot 4}$

Step 3.1.10

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

$x = \frac{32 \pm 12 \sqrt{y (- y + 4)}}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$x = \frac{32 \pm 12 \sqrt{y (- y + 4)}}{8}$

Step 3.3

Simplify $\frac{32 \pm 12 \sqrt{y (- y + 4)}}{8}$ .

$x = \frac{8 \pm 3 \sqrt{y (- y + 4)}}{2}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{8 + 3 \sqrt{y (- y + 4)}}{2}$

$x = \frac{8 - 3 \sqrt{y (- y + 4)}}{2}$