Finite Math Examples

$p (x) = - x^{2} + 62 x + 77$

Step 1

Set $- x^{2} + 62 x + 77$ equal to $0$ .

$- x^{2} + 62 x + 77 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = - 1$ , $b = 62$ , and $c = 77$ into the quadratic formula and solve for $x$ .

$\frac{- 62 \pm \sqrt{62^{2} - 4 \cdot (- 1 \cdot 77)}}{2 \cdot - 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $62$ to the power of $2$ .

$x = \frac{- 62 \pm \sqrt{3844 - 4 \cdot - 1 \cdot 77}}{2 \cdot - 1}$

Step 2.3.1.2

Multiply $- 4 \cdot - 1 \cdot 77$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 62 \pm \sqrt{3844 + 4 \cdot 77}}{2 \cdot - 1}$

Step 2.3.1.2.2

Multiply $4$ by $77$ .

$x = \frac{- 62 \pm \sqrt{3844 + 308}}{2 \cdot - 1}$

Step 2.3.1.3

Add $3844$ and $308$ .

$x = \frac{- 62 \pm \sqrt{4152}}{2 \cdot - 1}$

Step 2.3.1.4

Rewrite $4152$ as $2^{2} \cdot 1038$ .

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Step 2.3.1.4.1

Factor $4$ out of $4152$ .

$x = \frac{- 62 \pm \sqrt{4 (1038)}}{2 \cdot - 1}$

Step 2.3.1.4.2

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

$x = \frac{- 62 \pm \sqrt{2^{2} \cdot 1038}}{2 \cdot - 1}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 62 \pm 2 \sqrt{1038}}{2 \cdot - 1}$

Step 2.3.2

Multiply $2$ by $- 1$ .

$x = \frac{- 62 \pm 2 \sqrt{1038}}{- 2}$

Step 2.3.3

Simplify $\frac{- 62 \pm 2 \sqrt{1038}}{- 2}$ .

$x = 31 \pm \sqrt{1038}$

Step 2.4

The final answer is the combination of both solutions.

$x = 31 + \sqrt{1038}, 31 - \sqrt{1038}$

$x = 31 \pm \sqrt{1038}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 31 \pm \sqrt{1038}$

Decimal Form:

$x = 63.21800738 \dots, - 1.21800738 \dots$

Step 4