Evaluate Using the Given Value 12 , 250=-0.1x^2+70x

$12$ , $250 = - 0.1 x^{2} + 70 x$

Step 1

Rewrite the equation as $- 0.1 x^{2} + 70 x = 250$ .

$- 0.1 x^{2} + 70 x = 250$

Step 2

Subtract $250$ from both sides of the equation.

$- 0.1 x^{2} + 70 x - 250 = 0$

Step 3

Factor $- 0.1$ out of $- 0.1 x^{2} + 70 x - 250$ .

Factor $- 0.1$ out of $- 0.1 x^{2}$ .

$- 0.1 x^{2} + 70 x - 250 = 0$

Factor $- 0.1$ out of $70 x$ .

$- 0.1 x^{2} - 0.1 (- 700 x) - 250 = 0$

Factor $- 0.1$ out of $- 250$ .

$- 0.1 x^{2} - 0.1 (- 700 x) - 0.1 \cdot 2500 = 0$

Factor $- 0.1$ out of $- 0.1 (x^{2}) - 0.1 (- 700 x)$ .

$- 0.1 (x^{2} - 700 x) - 0.1 \cdot 2500 = 0$

Factor $- 0.1$ out of $- 0.1 (x^{2} - 700 x) - 0.1 (2500)$ .

$- 0.1 (x^{2} - 700 x + 2500) = 0$

Step 4

Divide each term in $- 0.1 (x^{2} - 700 x + 2500) = 0$ by $- 0.1$ and simplify.

Divide each term in $- 0.1 (x^{2} - 700 x + 2500) = 0$ by $- 0.1$ .

$\frac{- 0.1 (x^{2} - 700 x + 2500)}{- 0.1} = \frac{0}{- 0.1}$

Simplify the left side.

Cancel the common factor of $- 0.1$ .

Cancel the common factor.

$\frac{- 0.1 (x^{2} - 700 x + 2500)}{- 0.1} = \frac{0}{- 0.1}$

Divide $x^{2} - 700 x + 2500$ by $1$ .

$x^{2} - 700 x + 2500 = \frac{0}{- 0.1}$

Simplify the right side.

Divide $0$ by $- 0.1$ .

$x^{2} - 700 x + 2500 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 700$ , and $c = 2500$ into the quadratic formula and solve for $x$ .

$\frac{700 \pm \sqrt{{(- 700)}^{2} - 4 \cdot (1 \cdot 2500)}}{2 \cdot 1}$

Step 7

Simplify.

Simplify the numerator.

Raise $- 700$ to the power of $2$ .

$x = \frac{700 \pm \sqrt{490000 - 4 \cdot 1 \cdot 2500}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 2500$ .

Multiply $- 4$ by $1$ .

$x = \frac{700 \pm \sqrt{490000 - 4 \cdot 2500}}{2 \cdot 1}$

Multiply $- 4$ by $2500$ .

$x = \frac{700 \pm \sqrt{490000 - 10000}}{2 \cdot 1}$

Subtract $10000$ from $490000$ .

$x = \frac{700 \pm \sqrt{480000}}{2 \cdot 1}$

Rewrite $480000$ as $400^{2} \cdot 3$ .

Factor $160000$ out of $480000$ .

$x = \frac{700 \pm \sqrt{160000 (3)}}{2 \cdot 1}$

Rewrite $160000$ as $400^{2}$ .

$x = \frac{700 \pm \sqrt{400^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{700 \pm 400 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{700 \pm 400 \sqrt{3}}{2}$

Simplify $\frac{700 \pm 400 \sqrt{3}}{2}$ .

$x = 350 \pm 200 \sqrt{3}$

Step 8

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

Simplify the numerator.

Raise $- 700$ to the power of $2$ .

$x = \frac{700 \pm \sqrt{490000 - 4 \cdot 1 \cdot 2500}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 2500$ .

Multiply $- 4$ by $1$ .

$x = \frac{700 \pm \sqrt{490000 - 4 \cdot 2500}}{2 \cdot 1}$

Multiply $- 4$ by $2500$ .

$x = \frac{700 \pm \sqrt{490000 - 10000}}{2 \cdot 1}$

Subtract $10000$ from $490000$ .

$x = \frac{700 \pm \sqrt{480000}}{2 \cdot 1}$

Rewrite $480000$ as $400^{2} \cdot 3$ .

Factor $160000$ out of $480000$ .

$x = \frac{700 \pm \sqrt{160000 (3)}}{2 \cdot 1}$

Rewrite $160000$ as $400^{2}$ .

$x = \frac{700 \pm \sqrt{400^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{700 \pm 400 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{700 \pm 400 \sqrt{3}}{2}$

Simplify $\frac{700 \pm 400 \sqrt{3}}{2}$ .

$x = 350 \pm 200 \sqrt{3}$

Change the $\pm$ to $+$ .

$x = 350 + 200 \sqrt{3}$

Step 9

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

Simplify the numerator.

Raise $- 700$ to the power of $2$ .

$x = \frac{700 \pm \sqrt{490000 - 4 \cdot 1 \cdot 2500}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 2500$ .

Multiply $- 4$ by $1$ .

$x = \frac{700 \pm \sqrt{490000 - 4 \cdot 2500}}{2 \cdot 1}$

Multiply $- 4$ by $2500$ .

$x = \frac{700 \pm \sqrt{490000 - 10000}}{2 \cdot 1}$

Subtract $10000$ from $490000$ .

$x = \frac{700 \pm \sqrt{480000}}{2 \cdot 1}$

Rewrite $480000$ as $400^{2} \cdot 3$ .

Factor $160000$ out of $480000$ .

$x = \frac{700 \pm \sqrt{160000 (3)}}{2 \cdot 1}$

Rewrite $160000$ as $400^{2}$ .

$x = \frac{700 \pm \sqrt{400^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{700 \pm 400 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{700 \pm 400 \sqrt{3}}{2}$

Simplify $\frac{700 \pm 400 \sqrt{3}}{2}$ .

$x = 350 \pm 200 \sqrt{3}$

Change the $\pm$ to $-$ .

$x = 350 - 200 \sqrt{3}$

Step 10

The final answer is the combination of both solutions.

$x = 350 + 200 \sqrt{3}, 350 - 200 \sqrt{3}$

Step 11

Replace the variable $x$ with $350 + 200 \sqrt{3}$ in the expression.

$12$

Evaluate Using the Given Value 12 , 250=-0.1x^2+70x

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