Finite Math Examples

$7 = 0.0005 x^{2} - 0.16 x + 8$

Step 1

Rewrite the equation as $0.0005 x^{2} - 0.16 x + 8 = 7$ .

$0.0005 x^{2} - 0.16 x + 8 = 7$

Step 2

Subtract $7$ from both sides of the equation.

$0.0005 x^{2} - 0.16 x + 8 - 7 = 0$

Step 3

Subtract $7$ from $8$ .

$0.0005 x^{2} - 0.16 x + 1 = 0$

Step 4

Factor $0.0005$ out of $0.0005 x^{2} - 0.16 x + 1$ .

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Step 4.1

Factor $0.0005$ out of $0.0005 x^{2}$ .

$0.0005 (x^{2}) - 0.16 x + 1 = 0$

Step 4.2

Factor $0.0005$ out of $- 0.16 x$ .

$0.0005 (x^{2}) + 0.0005 (- 320 x) + 1 = 0$

Step 4.3

Factor $0.0005$ out of $1$ .

$0.0005 x^{2} + 0.0005 (- 320 x) + 0.0005 \cdot 2000 = 0$

Step 4.4

Factor $0.0005$ out of $0.0005 x^{2} + 0.0005 (- 320 x)$ .

$0.0005 (x^{2} - 320 x) + 0.0005 \cdot 2000 = 0$

Step 4.5

Factor $0.0005$ out of $0.0005 (x^{2} - 320 x) + 0.0005 \cdot 2000$ .

$0.0005 (x^{2} - 320 x + 2000) = 0$

Step 5

Divide each term in $0.0005 (x^{2} - 320 x + 2000) = 0$ by $0.0005$ and simplify.

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Step 5.1

Divide each term in $0.0005 (x^{2} - 320 x + 2000) = 0$ by $0.0005$ .

$\frac{0.0005 (x^{2} - 320 x + 2000)}{0.0005} = \frac{0}{0.0005}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.0005$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.0005 (x^{2} - 320 x + 2000)}{0.0005} = \frac{0}{0.0005}$

Step 5.2.1.2

Divide $x^{2} - 320 x + 2000$ by $1$ .

$x^{2} - 320 x + 2000 = \frac{0}{0.0005}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $0.0005$ .

$x^{2} - 320 x + 2000 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = - 320$ , and $c = 2000$ into the quadratic formula and solve for $x$ .

$\frac{320 \pm \sqrt{{(- 320)}^{2} - 4 \cdot (1 \cdot 2000)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 320$ to the power of $2$ .

$x = \frac{320 \pm \sqrt{102400 - 4 \cdot 1 \cdot 2000}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 2000$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{320 \pm \sqrt{102400 - 4 \cdot 2000}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $2000$ .

$x = \frac{320 \pm \sqrt{102400 - 8000}}{2 \cdot 1}$

Step 8.1.3

Subtract $8000$ from $102400$ .

$x = \frac{320 \pm \sqrt{94400}}{2 \cdot 1}$

Step 8.1.4

Rewrite $94400$ as $40^{2} \cdot 59$ .

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Step 8.1.4.1

Factor $1600$ out of $94400$ .

$x = \frac{320 \pm \sqrt{1600 (59)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $1600$ as $40^{2}$ .

$x = \frac{320 \pm \sqrt{40^{2} \cdot 59}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{320 \pm 40 \sqrt{59}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{320 \pm 40 \sqrt{59}}{2}$

Step 8.3

Simplify $\frac{320 \pm 40 \sqrt{59}}{2}$ .

$x = 160 \pm 20 \sqrt{59}$

Step 9

The final answer is the combination of both solutions.

$x = 160 + 20 \sqrt{59}, 160 - 20 \sqrt{59}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 160 + 20 \sqrt{59}, 160 - 20 \sqrt{59}$

Decimal Form:

$x = 313.62291495 \dots, 6.37708504 \dots$