Finite Math Examples

$x^{2} + 2 x + y^{2} + 3 y - 12 = 0$

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 $a = 1$ , $b = 2$ , and $c = y^{2} + 3 y - 12$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot (y^{2} + 3 y - 12))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot (y^{2} + 3 y - 12)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot (y^{2} + 3 y - 12)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 4 (3 y) - 4 \cdot - 12}}{2 \cdot 1}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $3$ by $- 4$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 12 y - 4 \cdot - 12}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $- 4$ by $- 12$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 12 y + 48}}{2 \cdot 1}$

Step 3.1.5

Add $4$ and $48$ .

$x = \frac{- 2 \pm \sqrt{- 4 y^{2} - 12 y + 52}}{2 \cdot 1}$

Step 3.1.6

Factor $4$ out of $- 4 y^{2} - 12 y + 52$ .

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Step 3.1.6.1

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

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) - 12 y + 52}}{2 \cdot 1}$

Step 3.1.6.2

Factor $4$ out of $- 12 y$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) + 4 (- 3 y) + 52}}{2 \cdot 1}$

Step 3.1.6.3

Factor $4$ out of $52$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) + 4 (- 3 y) + 4 (13)}}{2 \cdot 1}$

Step 3.1.6.4

Factor $4$ out of $4 (- y^{2}) + 4 (- 3 y)$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2} - 3 y) + 4 (13)}}{2 \cdot 1}$

Step 3.1.6.5

Factor $4$ out of $4 (- y^{2} - 3 y) + 4 (13)$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2} - 3 y + 13)}}{2 \cdot 1}$

Step 3.1.7

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

$x = \frac{- 2 \pm \sqrt{2^{2} (- y^{2} - 3 y + 13)}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2}$

Step 3.3

Simplify $\frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2}$ .

$x = - 1 \pm \sqrt{- y^{2} - 3 y + 13}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot (y^{2} + 3 y - 12)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot (y^{2} + 3 y - 12)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 4 (3 y) - 4 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $3$ by $- 4$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 12 y - 4 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 4$ by $- 12$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 12 y + 48}}{2 \cdot 1}$

Step 4.1.5

Add $4$ and $48$ .

$x = \frac{- 2 \pm \sqrt{- 4 y^{2} - 12 y + 52}}{2 \cdot 1}$

Step 4.1.6

Factor $4$ out of $- 4 y^{2} - 12 y + 52$ .

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Step 4.1.6.1

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

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) - 12 y + 52}}{2 \cdot 1}$

Step 4.1.6.2

Factor $4$ out of $- 12 y$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) + 4 (- 3 y) + 52}}{2 \cdot 1}$

Step 4.1.6.3

Factor $4$ out of $52$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) + 4 (- 3 y) + 4 (13)}}{2 \cdot 1}$

Step 4.1.6.4

Factor $4$ out of $4 (- y^{2}) + 4 (- 3 y)$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2} - 3 y) + 4 (13)}}{2 \cdot 1}$

Step 4.1.6.5

Factor $4$ out of $4 (- y^{2} - 3 y) + 4 (13)$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2} - 3 y + 13)}}{2 \cdot 1}$

Step 4.1.7

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

$x = \frac{- 2 \pm \sqrt{2^{2} (- y^{2} - 3 y + 13)}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2}$

Step 4.3

Simplify $\frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2}$ .

$x = - 1 \pm \sqrt{- y^{2} - 3 y + 13}$

Step 4.4

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{- y^{2} - 3 y + 13}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot (y^{2} + 3 y - 12)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot (y^{2} + 3 y - 12)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 4 (3 y) - 4 \cdot - 12}}{2 \cdot 1}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $3$ by $- 4$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 12 y - 4 \cdot - 12}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $- 4$ by $- 12$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 y^{2} - 12 y + 48}}{2 \cdot 1}$

Step 5.1.5

Add $4$ and $48$ .

$x = \frac{- 2 \pm \sqrt{- 4 y^{2} - 12 y + 52}}{2 \cdot 1}$

Step 5.1.6

Factor $4$ out of $- 4 y^{2} - 12 y + 52$ .

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Step 5.1.6.1

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

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) - 12 y + 52}}{2 \cdot 1}$

Step 5.1.6.2

Factor $4$ out of $- 12 y$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) + 4 (- 3 y) + 52}}{2 \cdot 1}$

Step 5.1.6.3

Factor $4$ out of $52$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2}) + 4 (- 3 y) + 4 (13)}}{2 \cdot 1}$

Step 5.1.6.4

Factor $4$ out of $4 (- y^{2}) + 4 (- 3 y)$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2} - 3 y) + 4 (13)}}{2 \cdot 1}$

Step 5.1.6.5

Factor $4$ out of $4 (- y^{2} - 3 y) + 4 (13)$ .

$x = \frac{- 2 \pm \sqrt{4 (- y^{2} - 3 y + 13)}}{2 \cdot 1}$

Step 5.1.7

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

$x = \frac{- 2 \pm \sqrt{2^{2} (- y^{2} - 3 y + 13)}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2}$

Step 5.3

Simplify $\frac{- 2 \pm 2 \sqrt{- y^{2} - 3 y + 13}}{2}$ .

$x = - 1 \pm \sqrt{- y^{2} - 3 y + 13}$

Step 5.4

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{- y^{2} - 3 y + 13}$

Step 6

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{- y^{2} - 3 y + 13}$

$x = - 1 - \sqrt{- y^{2} - 3 y + 13}$