Pre-Algebra Examples

$- \frac{1}{2} \cdot (32 x^{2}) + 1000 x + 50 = 0$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $2$ .

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Step 1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} \cdot (32 x^{2}) + 1000 x + 50 = 0$

Step 1.1.2

Factor $2$ out of $32 x^{2}$ .

$\frac{- 1}{2} \cdot (2 (16 x^{2})) + 1000 x + 50 = 0$

Step 1.1.3

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (16 x^{2})) + 1000 x + 50 = 0$

Step 1.1.4

Rewrite the expression.

$- 1 \cdot (16 x^{2}) + 1000 x + 50 = 0$

Step 1.2

Multiply $16$ by $- 1$ .

$- 16 x^{2} + 1000 x + 50 = 0$

Step 2

Factor $- 2$ out of $- 16 x^{2} + 1000 x + 50$ .

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Step 2.1

Factor $- 2$ out of $- 16 x^{2}$ .

$- 2 (8 x^{2}) + 1000 x + 50 = 0$

Step 2.2

Factor $- 2$ out of $1000 x$ .

$- 2 (8 x^{2}) - 2 (- 500 x) + 50 = 0$

Step 2.3

Factor $- 2$ out of $50$ .

$- 2 (8 x^{2}) - 2 (- 500 x) - 2 \cdot - 25 = 0$

Step 2.4

Factor $- 2$ out of $- 2 (8 x^{2}) - 2 (- 500 x)$ .

$- 2 (8 x^{2} - 500 x) - 2 \cdot - 25 = 0$

Step 2.5

Factor $- 2$ out of $- 2 (8 x^{2} - 500 x) - 2 (- 25)$ .

$- 2 (8 x^{2} - 500 x - 25) = 0$

Step 3

Divide each term in $- 2 (8 x^{2} - 500 x - 25) = 0$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 (8 x^{2} - 500 x - 25) = 0$ by $- 2$ .

$\frac{- 2 (8 x^{2} - 500 x - 25)}{- 2} = \frac{0}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 (8 x^{2} - 500 x - 25)}{- 2} = \frac{0}{- 2}$

Step 3.2.1.2

Divide $8 x^{2} - 500 x - 25$ by $1$ .

$8 x^{2} - 500 x - 25 = \frac{0}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 2$ .

$8 x^{2} - 500 x - 25 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 8$ , $b = - 500$ , and $c = - 25$ into the quadratic formula and solve for $x$ .

$\frac{500 \pm \sqrt{{(- 500)}^{2} - 4 \cdot (8 \cdot - 25)}}{2 \cdot 8}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 500$ to the power of $2$ .

$x = \frac{500 \pm \sqrt{250000 - 4 \cdot 8 \cdot - 25}}{2 \cdot 8}$

Step 6.1.2

Multiply $- 4 \cdot 8 \cdot - 25$ .

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Step 6.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{500 \pm \sqrt{250000 - 32 \cdot - 25}}{2 \cdot 8}$

Step 6.1.2.2

Multiply $- 32$ by $- 25$ .

$x = \frac{500 \pm \sqrt{250000 + 800}}{2 \cdot 8}$

Step 6.1.3

Add $250000$ and $800$ .

$x = \frac{500 \pm \sqrt{250800}}{2 \cdot 8}$

Step 6.1.4

Rewrite $250800$ as $20^{2} \cdot 627$ .

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Step 6.1.4.1

Factor $400$ out of $250800$ .

$x = \frac{500 \pm \sqrt{400 (627)}}{2 \cdot 8}$

Step 6.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{500 \pm \sqrt{20^{2} \cdot 627}}{2 \cdot 8}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{500 \pm 20 \sqrt{627}}{2 \cdot 8}$

Step 6.2

Multiply $2$ by $8$ .

$x = \frac{500 \pm 20 \sqrt{627}}{16}$

Step 6.3

Simplify $\frac{500 \pm 20 \sqrt{627}}{16}$ .

$x = \frac{125 \pm 5 \sqrt{627}}{4}$

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 $- 500$ to the power of $2$ .

$x = \frac{500 \pm \sqrt{250000 - 4 \cdot 8 \cdot - 25}}{2 \cdot 8}$

Step 7.1.2

Multiply $- 4 \cdot 8 \cdot - 25$ .

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Step 7.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{500 \pm \sqrt{250000 - 32 \cdot - 25}}{2 \cdot 8}$

Step 7.1.2.2

Multiply $- 32$ by $- 25$ .

$x = \frac{500 \pm \sqrt{250000 + 800}}{2 \cdot 8}$

Step 7.1.3

Add $250000$ and $800$ .

$x = \frac{500 \pm \sqrt{250800}}{2 \cdot 8}$

Step 7.1.4

Rewrite $250800$ as $20^{2} \cdot 627$ .

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Step 7.1.4.1

Factor $400$ out of $250800$ .

$x = \frac{500 \pm \sqrt{400 (627)}}{2 \cdot 8}$

Step 7.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{500 \pm \sqrt{20^{2} \cdot 627}}{2 \cdot 8}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{500 \pm 20 \sqrt{627}}{2 \cdot 8}$

Step 7.2

Multiply $2$ by $8$ .

$x = \frac{500 \pm 20 \sqrt{627}}{16}$

Step 7.3

Simplify $\frac{500 \pm 20 \sqrt{627}}{16}$ .

$x = \frac{125 \pm 5 \sqrt{627}}{4}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{125 + 5 \sqrt{627}}{4}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 500$ to the power of $2$ .

$x = \frac{500 \pm \sqrt{250000 - 4 \cdot 8 \cdot - 25}}{2 \cdot 8}$

Step 8.1.2

Multiply $- 4 \cdot 8 \cdot - 25$ .

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Step 8.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{500 \pm \sqrt{250000 - 32 \cdot - 25}}{2 \cdot 8}$

Step 8.1.2.2

Multiply $- 32$ by $- 25$ .

$x = \frac{500 \pm \sqrt{250000 + 800}}{2 \cdot 8}$

Step 8.1.3

Add $250000$ and $800$ .

$x = \frac{500 \pm \sqrt{250800}}{2 \cdot 8}$

Step 8.1.4

Rewrite $250800$ as $20^{2} \cdot 627$ .

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Step 8.1.4.1

Factor $400$ out of $250800$ .

$x = \frac{500 \pm \sqrt{400 (627)}}{2 \cdot 8}$

Step 8.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{500 \pm \sqrt{20^{2} \cdot 627}}{2 \cdot 8}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{500 \pm 20 \sqrt{627}}{2 \cdot 8}$

Step 8.2

Multiply $2$ by $8$ .

$x = \frac{500 \pm 20 \sqrt{627}}{16}$

Step 8.3

Simplify $\frac{500 \pm 20 \sqrt{627}}{16}$ .

$x = \frac{125 \pm 5 \sqrt{627}}{4}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{125 - 5 \sqrt{627}}{4}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{125 + 5 \sqrt{627}}{4}, \frac{125 - 5 \sqrt{627}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{125 + 5 \sqrt{627}}{4}, \frac{125 - 5 \sqrt{627}}{4}$

Decimal Form:

$x = 62.54996006 \dots, - 0.04996006 \dots$