Pre-Algebra Examples

$10 = - 5 x^{2} + 23 x + 23$

Step 1

Rewrite the equation as $- 5 x^{2} + 23 x + 23 = 10$ .

$- 5 x^{2} + 23 x + 23 = 10$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $10$ from both sides of the equation.

$- 5 x^{2} + 23 x + 23 - 10 = 0$

Step 2.2

Subtract $10$ from $23$ .

$- 5 x^{2} + 23 x + 13 = 0$

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 $a = - 5$ , $b = 23$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{- 23 \pm \sqrt{23^{2} - 4 \cdot (- 5 \cdot 13)}}{2 \cdot - 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot - 5 \cdot 13}}{2 \cdot - 5}$

Step 5.1.2

Multiply $- 4 \cdot - 5 \cdot 13$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 23 \pm \sqrt{529 + 20 \cdot 13}}{2 \cdot - 5}$

Step 5.1.2.2

Multiply $20$ by $13$ .

$x = \frac{- 23 \pm \sqrt{529 + 260}}{2 \cdot - 5}$

Step 5.1.3

Add $529$ and $260$ .

$x = \frac{- 23 \pm \sqrt{789}}{2 \cdot - 5}$

Step 5.2

Multiply $2$ by $- 5$ .

$x = \frac{- 23 \pm \sqrt{789}}{- 10}$

Step 5.3

Simplify $\frac{- 23 \pm \sqrt{789}}{- 10}$ .

$x = \frac{23 \pm \sqrt{789}}{10}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot - 5 \cdot 13}}{2 \cdot - 5}$

Step 6.1.2

Multiply $- 4 \cdot - 5 \cdot 13$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 23 \pm \sqrt{529 + 20 \cdot 13}}{2 \cdot - 5}$

Step 6.1.2.2

Multiply $20$ by $13$ .

$x = \frac{- 23 \pm \sqrt{529 + 260}}{2 \cdot - 5}$

Step 6.1.3

Add $529$ and $260$ .

$x = \frac{- 23 \pm \sqrt{789}}{2 \cdot - 5}$

Step 6.2

Multiply $2$ by $- 5$ .

$x = \frac{- 23 \pm \sqrt{789}}{- 10}$

Step 6.3

Simplify $\frac{- 23 \pm \sqrt{789}}{- 10}$ .

$x = \frac{23 \pm \sqrt{789}}{10}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{23 + \sqrt{789}}{10}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot - 5 \cdot 13}}{2 \cdot - 5}$

Step 7.1.2

Multiply $- 4 \cdot - 5 \cdot 13$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 23 \pm \sqrt{529 + 20 \cdot 13}}{2 \cdot - 5}$

Step 7.1.2.2

Multiply $20$ by $13$ .

$x = \frac{- 23 \pm \sqrt{529 + 260}}{2 \cdot - 5}$

Step 7.1.3

Add $529$ and $260$ .

$x = \frac{- 23 \pm \sqrt{789}}{2 \cdot - 5}$

Step 7.2

Multiply $2$ by $- 5$ .

$x = \frac{- 23 \pm \sqrt{789}}{- 10}$

Step 7.3

Simplify $\frac{- 23 \pm \sqrt{789}}{- 10}$ .

$x = \frac{23 \pm \sqrt{789}}{10}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{23 - \sqrt{789}}{10}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{23 + \sqrt{789}}{10}, \frac{23 - \sqrt{789}}{10}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{23 + \sqrt{789}}{10}, \frac{23 - \sqrt{789}}{10}$

Decimal Form:

$x = 5.10891438 \dots, - 0.50891438 \dots$