Finite Math Examples

$x^{2} + y^{2} - 6 x + 4 y = - 11$

Step 1

Add $11$ to both sides of the equation.

$x^{2} + y^{2} - 6 x + 4 y + 11 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 6$ , and $c = y^{2} + 4 y + 11$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot (y^{2} + 4 y + 11))}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (y^{2} + 4 y + 11)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot (y^{2} + 4 y + 11)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 4 (4 y) - 4 \cdot 11}}{2 \cdot 1}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $4$ by $- 4$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 16 y - 4 \cdot 11}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 4$ by $11$ .

$x = \frac{6 \pm \sqrt{36 - 4 y^{2} - 16 y - 44}}{2 \cdot 1}$

Step 4.1.5

Subtract $44$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4 y^{2} - 16 y - 8}}{2 \cdot 1}$

Step 4.1.6

Factor $4$ out of $- 4 y^{2} - 16 y - 8$ .

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Step 4.1.6.1

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

$x = \frac{6 \pm \sqrt{4 (- y^{2}) - 16 y - 8}}{2 \cdot 1}$

Step 4.1.6.2

Factor $4$ out of $- 16 y$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (- 4 y) - 8}}{2 \cdot 1}$

Step 4.1.6.3

Factor $4$ out of $- 8$ .

$x = \frac{6 \pm \sqrt{4 (- y^{2}) + 4 (- 4 y) + 4 (- 2)}}{2 \cdot 1}$

Step 4.1.6.4

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

$x = \frac{6 \pm \sqrt{4 (- y^{2} - 4 y) + 4 (- 2)}}{2 \cdot 1}$

Step 4.1.6.5

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

$x = \frac{6 \pm \sqrt{4 (- y^{2} - 4 y - 2)}}{2 \cdot 1}$

Step 4.1.7

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

$x = \frac{6 \pm \sqrt{2^{2} (- y^{2} - 4 y - 2)}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{- y^{2} - 4 y - 2}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{- y^{2} - 4 y - 2}}{2}$

Step 4.3

Simplify $\frac{6 \pm 2 \sqrt{- y^{2} - 4 y - 2}}{2}$ .

$x = 3 \pm \sqrt{- y^{2} - 4 y - 2}$

Step 5

The final answer is the combination of both solutions.

$x = 3 + \sqrt{- y^{2} - 4 y - 2}$

$x = 3 - \sqrt{- y^{2} - 4 y - 2}$