Finite Math Examples

$x^{2} - 12 x + y + 40 = 0$

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 $a = 1$ , $b = - 12$ , and $c = y + 40$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot (y + 40))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot (y + 40)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot (y + 40)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$x = \frac{12 \pm \sqrt{144 - 4 y - 4 \cdot 40}}{2 \cdot 1}$

Step 3.1.4

Multiply $- 4$ by $40$ .

$x = \frac{12 \pm \sqrt{144 - 4 y - 160}}{2 \cdot 1}$

Step 3.1.5

Subtract $160$ from $144$ .

$x = \frac{12 \pm \sqrt{- 4 y - 16}}{2 \cdot 1}$

Step 3.1.6

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

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Step 3.1.6.1

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

$x = \frac{12 \pm \sqrt{4 (- y) - 16}}{2 \cdot 1}$

Step 3.1.6.2

Factor $4$ out of $- 16$ .

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

Step 3.1.6.3

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

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

Step 3.1.7

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

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

Step 3.1.8

Pull terms out from under the radical.

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

Step 3.2

Multiply $2$ by $1$ .

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

Step 3.3

Simplify $\frac{12 \pm 2 \sqrt{- y - 4}}{2}$ .

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

Step 4

The final answer is the combination of both solutions.

$x = 6 + \sqrt{- y - 4}$

$x = 6 - \sqrt{- y - 4}$