Finite Math Examples

$2 x^{2} + 4 y^{2} - 24 x + 32 y + 128 = 0$

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 $a = 2$ , $b = - 24$ , and $c = 4 y^{2} + 32 y + 128$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (2 \cdot (4 y^{2} + 32 y + 128))}}{2 \cdot 2}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 2 \cdot (4 y^{2} + 32 y + 128)}}{2 \cdot 2}$

Step 3.1.2

Multiply $- 4$ by $2$ .

$x = \frac{24 \pm \sqrt{576 - 8 \cdot (4 y^{2} + 32 y + 128)}}{2 \cdot 2}$

Step 3.1.3

Apply the distributive property.

$x = \frac{24 \pm \sqrt{576 - 8 (4 y^{2}) - 8 (32 y) - 8 \cdot 128}}{2 \cdot 2}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $4$ by $- 8$ .

$x = \frac{24 \pm \sqrt{576 - 32 y^{2} - 8 (32 y) - 8 \cdot 128}}{2 \cdot 2}$

Step 3.1.4.2

Multiply $32$ by $- 8$ .

$x = \frac{24 \pm \sqrt{576 - 32 y^{2} - 256 y - 8 \cdot 128}}{2 \cdot 2}$

Step 3.1.4.3

Multiply $- 8$ by $128$ .

$x = \frac{24 \pm \sqrt{576 - 32 y^{2} - 256 y - 1024}}{2 \cdot 2}$

Step 3.1.5

Subtract $1024$ from $576$ .

$x = \frac{24 \pm \sqrt{- 32 y^{2} - 256 y - 448}}{2 \cdot 2}$

Step 3.1.6

Factor $32$ out of $- 32 y^{2} - 256 y - 448$ .

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Step 3.1.6.1

Factor $32$ out of $- 32 y^{2}$ .

$x = \frac{24 \pm \sqrt{32 (- y^{2}) - 256 y - 448}}{2 \cdot 2}$

Step 3.1.6.2

Factor $32$ out of $- 256 y$ .

$x = \frac{24 \pm \sqrt{32 (- y^{2}) + 32 (- 8 y) - 448}}{2 \cdot 2}$

Step 3.1.6.3

Factor $32$ out of $- 448$ .

$x = \frac{24 \pm \sqrt{32 (- y^{2}) + 32 (- 8 y) + 32 (- 14)}}{2 \cdot 2}$

Step 3.1.6.4

Factor $32$ out of $32 (- y^{2}) + 32 (- 8 y)$ .

$x = \frac{24 \pm \sqrt{32 (- y^{2} - 8 y) + 32 (- 14)}}{2 \cdot 2}$

Step 3.1.6.5

Factor $32$ out of $32 (- y^{2} - 8 y) + 32 (- 14)$ .

$x = \frac{24 \pm \sqrt{32 (- y^{2} - 8 y - 14)}}{2 \cdot 2}$

Step 3.1.7

Rewrite $32 (- y^{2} - 8 y - 14)$ as ${(2^{2})}^{2} \cdot (2 (- y^{2} - 8 y - 14))$ .

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Step 3.1.7.1

Factor $16$ out of $32$ .

$x = \frac{24 \pm \sqrt{16 (2) (- y^{2} - 8 y - 14)}}{2 \cdot 2}$

Step 3.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{24 \pm \sqrt{4^{2} \cdot (2 (- y^{2} - 8 y - 14))}}{2 \cdot 2}$

Step 3.1.7.3

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

$x = \frac{24 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- y^{2} - 8 y - 14))}}{2 \cdot 2}$

Step 3.1.7.4

Add parentheses.

$x = \frac{24 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- y^{2} - 8 y - 14))}}{2 \cdot 2}$

Step 3.1.8

Pull terms out from under the radical.

$x = \frac{24 \pm 2^{2} \sqrt{2 (- y^{2} - 8 y - 14)}}{2 \cdot 2}$

Step 3.1.9

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

$x = \frac{24 \pm 4 \sqrt{2 (- y^{2} - 8 y - 14)}}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$x = \frac{24 \pm 4 \sqrt{2 (- y^{2} - 8 y - 14)}}{4}$

Step 3.3

Simplify $\frac{24 \pm 4 \sqrt{2 (- y^{2} - 8 y - 14)}}{4}$ .

$x = 6 \pm \sqrt{2 (- y^{2} - 8 y - 14)}$

Step 4

The final answer is the combination of both solutions.

$x = 6 + \sqrt{2 (- y^{2} - 8 y - 14)}$

$x = 6 - \sqrt{2 (- y^{2} - 8 y - 14)}$