Trigonometry Examples

$200 x - 2 x^{2} = - x^{2} + 4 x + 3750$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Add $x^{2}$ to both sides of the equation.

$200 x - 2 x^{2} + x^{2} = 4 x + 3750$

Step 1.2

Subtract $4 x$ from both sides of the equation.

$200 x - 2 x^{2} + x^{2} - 4 x = 3750$

Step 1.3

Subtract $4 x$ from $200 x$ .

$- 2 x^{2} + x^{2} + 196 x = 3750$

Step 1.4

Add $- 2 x^{2}$ and $x^{2}$ .

$- x^{2} + 196 x = 3750$

Step 2

Subtract $3750$ from both sides of the equation.

$- x^{2} + 196 x - 3750 = 0$

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 $a = - 1$ , $b = 196$ , and $c = - 3750$ into the quadratic formula and solve for $x$ .

$\frac{- 196 \pm \sqrt{196^{2} - 4 \cdot (- 1 \cdot - 3750)}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $196$ to the power of $2$ .

$x = \frac{- 196 \pm \sqrt{38416 - 4 \cdot - 1 \cdot - 3750}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot - 3750$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 196 \pm \sqrt{38416 + 4 \cdot - 3750}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 3750$ .

$x = \frac{- 196 \pm \sqrt{38416 - 15000}}{2 \cdot - 1}$

Step 5.1.3

Subtract $15000$ from $38416$ .

$x = \frac{- 196 \pm \sqrt{23416}}{2 \cdot - 1}$

Step 5.1.4

Rewrite $23416$ as $2^{2} \cdot 5854$ .

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Step 5.1.4.1

Factor $4$ out of $23416$ .

$x = \frac{- 196 \pm \sqrt{4 (5854)}}{2 \cdot - 1}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 196 \pm \sqrt{2^{2} \cdot 5854}}{2 \cdot - 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 196 \pm 2 \sqrt{5854}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 196 \pm 2 \sqrt{5854}}{- 2}$

Step 5.3

Simplify $\frac{- 196 \pm 2 \sqrt{5854}}{- 2}$ .

$x = 98 \pm \sqrt{5854}$

Step 6

The final answer is the combination of both solutions.

$x = 98 + \sqrt{5854}, 98 - \sqrt{5854}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 98 + \sqrt{5854}, 98 - \sqrt{5854}$

Decimal Form:

$x = 174.51143705 \dots, 21.48856294 \dots$