Pre-Algebra Examples

$0.25 x^{2} - 8 x - 6000 = 0$

Step 1

Factor $0.25$ out of $0.25 x^{2} - 8 x - 6000$ .

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Step 1.1

Factor $0.25$ out of $0.25 x^{2}$ .

$0.25 (x^{2}) - 8 x - 6000 = 0$

Step 1.2

Factor $0.25$ out of $- 8 x$ .

$0.25 (x^{2}) + 0.25 (- 32 x) - 6000 = 0$

Step 1.3

Factor $0.25$ out of $- 6000$ .

$0.25 x^{2} + 0.25 (- 32 x) + 0.25 \cdot - 24000 = 0$

Step 1.4

Factor $0.25$ out of $0.25 x^{2} + 0.25 (- 32 x)$ .

$0.25 (x^{2} - 32 x) + 0.25 \cdot - 24000 = 0$

Step 1.5

Factor $0.25$ out of $0.25 (x^{2} - 32 x) + 0.25 \cdot - 24000$ .

$0.25 (x^{2} - 32 x - 24000) = 0$

Step 2

Divide each term in $0.25 (x^{2} - 32 x - 24000) = 0$ by $0.25$ and simplify.

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Step 2.1

Divide each term in $0.25 (x^{2} - 32 x - 24000) = 0$ by $0.25$ .

$\frac{0.25 (x^{2} - 32 x - 24000)}{0.25} = \frac{0}{0.25}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $0.25$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{0.25 (x^{2} - 32 x - 24000)}{0.25} = \frac{0}{0.25}$

Step 2.2.1.2

Divide $x^{2} - 32 x - 24000$ by $1$ .

$x^{2} - 32 x - 24000 = \frac{0}{0.25}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $0.25$ .

$x^{2} - 32 x - 24000 = 0$

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 $a = 1$ , $b = - 32$ , and $c = - 24000$ into the quadratic formula and solve for $x$ .

$\frac{32 \pm \sqrt{{(- 32)}^{2} - 4 \cdot (1 \cdot - 24000)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 32$ to the power of $2$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot 1 \cdot - 24000}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 24000$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot - 24000}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 24000$ .

$x = \frac{32 \pm \sqrt{1024 + 96000}}{2 \cdot 1}$

Step 5.1.3

Add $1024$ and $96000$ .

$x = \frac{32 \pm \sqrt{97024}}{2 \cdot 1}$

Step 5.1.4

Rewrite $97024$ as $16^{2} \cdot 379$ .

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Step 5.1.4.1

Factor $256$ out of $97024$ .

$x = \frac{32 \pm \sqrt{256 (379)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite $256$ as $16^{2}$ .

$x = \frac{32 \pm \sqrt{16^{2} \cdot 379}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{32 \pm 16 \sqrt{379}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{32 \pm 16 \sqrt{379}}{2}$

Step 5.3

Simplify $\frac{32 \pm 16 \sqrt{379}}{2}$ .

$x = 16 \pm 8 \sqrt{379}$

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 $- 32$ to the power of $2$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot 1 \cdot - 24000}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 24000$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot - 24000}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 24000$ .

$x = \frac{32 \pm \sqrt{1024 + 96000}}{2 \cdot 1}$

Step 6.1.3

Add $1024$ and $96000$ .

$x = \frac{32 \pm \sqrt{97024}}{2 \cdot 1}$

Step 6.1.4

Rewrite $97024$ as $16^{2} \cdot 379$ .

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Step 6.1.4.1

Factor $256$ out of $97024$ .

$x = \frac{32 \pm \sqrt{256 (379)}}{2 \cdot 1}$

Step 6.1.4.2

Rewrite $256$ as $16^{2}$ .

$x = \frac{32 \pm \sqrt{16^{2} \cdot 379}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{32 \pm 16 \sqrt{379}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{32 \pm 16 \sqrt{379}}{2}$

Step 6.3

Simplify $\frac{32 \pm 16 \sqrt{379}}{2}$ .

$x = 16 \pm 8 \sqrt{379}$

Step 6.4

Change the $\pm$ to $+$ .

$x = 16 + 8 \sqrt{379}$

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 $- 32$ to the power of $2$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot 1 \cdot - 24000}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 24000$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{32 \pm \sqrt{1024 - 4 \cdot - 24000}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 24000$ .

$x = \frac{32 \pm \sqrt{1024 + 96000}}{2 \cdot 1}$

Step 7.1.3

Add $1024$ and $96000$ .

$x = \frac{32 \pm \sqrt{97024}}{2 \cdot 1}$

Step 7.1.4

Rewrite $97024$ as $16^{2} \cdot 379$ .

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Step 7.1.4.1

Factor $256$ out of $97024$ .

$x = \frac{32 \pm \sqrt{256 (379)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $256$ as $16^{2}$ .

$x = \frac{32 \pm \sqrt{16^{2} \cdot 379}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{32 \pm 16 \sqrt{379}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{32 \pm 16 \sqrt{379}}{2}$

Step 7.3

Simplify $\frac{32 \pm 16 \sqrt{379}}{2}$ .

$x = 16 \pm 8 \sqrt{379}$

Step 7.4

Change the $\pm$ to $-$ .

$x = 16 - 8 \sqrt{379}$

Step 8

The final answer is the combination of both solutions.

$x = 16 + 8 \sqrt{379}, 16 - 8 \sqrt{379}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 16 + 8 \sqrt{379}, 16 - 8 \sqrt{379}$

Decimal Form:

$x = 171.74337867 \dots, - 139.74337867 \dots$