Finite Math Examples

$s = x^{2} + y^{2} - 8 x + 10 y - 4$

Step 1

Rewrite the equation as $x^{2} + y^{2} - 8 x + 10 y - 4 = s$ .

$x^{2} + y^{2} - 8 x + 10 y - 4 = s$

Step 2

Subtract $s$ from both sides of the equation.

$x^{2} + y^{2} - 8 x + 10 y - 4 - s = 0$

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 $a = 1$ , $b = - 8$ , and $c = y^{2} + 10 y - 4 - s$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot (y^{2} + 10 y - 4 - s))}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (y^{2} + 10 y - 4 - s)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot (y^{2} + 10 y - 4 - s)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$x = \frac{8 \pm \sqrt{64 - 4 y^{2} - 4 (10 y) - 4 \cdot - 4 - 4 (- s)}}{2 \cdot 1}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $10$ by $- 4$ .

$x = \frac{8 \pm \sqrt{64 - 4 y^{2} - 40 y - 4 \cdot - 4 - 4 (- s)}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $- 4$ by $- 4$ .

$x = \frac{8 \pm \sqrt{64 - 4 y^{2} - 40 y + 16 - 4 (- s)}}{2 \cdot 1}$

Step 5.1.4.3

Multiply $- 1$ by $- 4$ .

$x = \frac{8 \pm \sqrt{64 - 4 y^{2} - 40 y + 16 + 4 s}}{2 \cdot 1}$

Step 5.1.5

Add $64$ and $16$ .

$x = \frac{8 \pm \sqrt{- 4 y^{2} - 40 y + 80 + 4 s}}{2 \cdot 1}$

Step 5.1.6

Factor $4$ out of $- 4 y^{2} - 40 y + 80 + 4 s$ .

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Step 5.1.6.1

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

$x = \frac{8 \pm \sqrt{4 (- y^{2}) - 40 y + 80 + 4 s}}{2 \cdot 1}$

Step 5.1.6.2

Factor $4$ out of $- 40 y$ .

$x = \frac{8 \pm \sqrt{4 (- y^{2}) + 4 (- 10 y) + 80 + 4 s}}{2 \cdot 1}$

Step 5.1.6.3

Factor $4$ out of $80$ .

$x = \frac{8 \pm \sqrt{4 (- y^{2}) + 4 (- 10 y) + 4 \cdot 20 + 4 s}}{2 \cdot 1}$

Step 5.1.6.4

Factor $4$ out of $4 (- y^{2}) + 4 (- 10 y)$ .

$x = \frac{8 \pm \sqrt{4 (- y^{2} - 10 y) + 4 \cdot 20 + 4 s}}{2 \cdot 1}$

Step 5.1.6.5

Factor $4$ out of $4 (- y^{2} - 10 y) + 4 \cdot 20$ .

$x = \frac{8 \pm \sqrt{4 (- y^{2} - 10 y + 20) + 4 s}}{2 \cdot 1}$

Step 5.1.6.6

Factor $4$ out of $4 (- y^{2} - 10 y + 20) + 4 s$ .

$x = \frac{8 \pm \sqrt{4 (- y^{2} - 10 y + 20 + s)}}{2 \cdot 1}$

Step 5.1.7

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

$x = \frac{8 \pm \sqrt{2^{2} (- y^{2} - 10 y + 20 + s)}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{- y^{2} - 10 y + 20 + s}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{- y^{2} - 10 y + 20 + s}}{2}$

Step 5.3

Simplify $\frac{8 \pm 2 \sqrt{- y^{2} - 10 y + 20 + s}}{2}$ .

$x = 4 \pm \sqrt{- y^{2} - 10 y + 20 + s}$

Step 6

The final answer is the combination of both solutions.

$x = 4 + \sqrt{- y^{2} - 10 y + 20 + s}$

$x = 4 - \sqrt{- y^{2} - 10 y + 20 + s}$