Finite Math Examples

$y = - x^{2} + 6 x - 7$

Step 1

Rewrite the equation as $- x^{2} + 6 x - 7 = y$ .

$- x^{2} + 6 x - 7 = y$

Step 2

Subtract $y$ from both sides of the equation.

$- x^{2} + 6 x - 7 - y = 0$

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 $a = - 1$ , $b = 6$ , and $c = - 7 - y$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 1 \cdot (- 7 - y))}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot (- 7 - y)}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot (- 7 - y)}}{2 \cdot - 1}$

Step 5.1.3

Apply the distributive property.

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 7 + 4 (- y)}}{2 \cdot - 1}$

Step 5.1.4

Multiply $4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 - 28 + 4 (- y)}}{2 \cdot - 1}$

Step 5.1.5

Multiply $- 1$ by $4$ .

$x = \frac{- 6 \pm \sqrt{36 - 28 - 4 y}}{2 \cdot - 1}$

Step 5.1.6

Subtract $28$ from $36$ .

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

Step 5.1.7

Factor $4$ out of $8 - 4 y$ .

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Step 5.1.7.1

Factor $4$ out of $8$ .

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

Step 5.1.7.2

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

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

Step 5.1.7.3

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

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

Step 5.1.8

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

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

Step 5.1.9

Pull terms out from under the radical.

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

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

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

Step 6

The final answer is the combination of both solutions.

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

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