Finite Math Examples

$200 x - 2 x^{2} = - x^{2} + 10 x + 8400$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

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

$200 x - 2 x^{2} + x^{2} = 10 x + 8400$

Step 1.1.2

Subtract $10 x$ from both sides of the equation.

$200 x - 2 x^{2} + x^{2} - 10 x = 8400$

Step 1.1.3

Subtract $8400$ from both sides of the equation.

$200 x - 2 x^{2} + x^{2} - 10 x - 8400 = 0$

Step 1.2

Simplify $200 x - 2 x^{2} + x^{2} - 10 x - 8400$ .

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Step 1.2.1

Subtract $10 x$ from $200 x$ .

$- 2 x^{2} + x^{2} + 190 x - 8400 = 0$

Step 1.2.2

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

$- x^{2} + 190 x - 8400 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 1$ , $b = 190$ , and $c = - 8400$ into the quadratic formula and solve for $x$ .

$\frac{- 190 \pm \sqrt{190^{2} - 4 \cdot (- 1 \cdot - 8400)}}{2 \cdot - 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $190$ to the power of $2$ .

$x = \frac{- 190 \pm \sqrt{36100 - 4 \cdot - 1 \cdot - 8400}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4 \cdot - 1 \cdot - 8400$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 190 \pm \sqrt{36100 + 4 \cdot - 8400}}{2 \cdot - 1}$

Step 4.1.2.2

Multiply $4$ by $- 8400$ .

$x = \frac{- 190 \pm \sqrt{36100 - 33600}}{2 \cdot - 1}$

Step 4.1.3

Subtract $33600$ from $36100$ .

$x = \frac{- 190 \pm \sqrt{2500}}{2 \cdot - 1}$

Step 4.1.4

Rewrite $2500$ as $50^{2}$ .

$x = \frac{- 190 \pm \sqrt{50^{2}}}{2 \cdot - 1}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 190 \pm 50}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$x = \frac{- 190 \pm 50}{- 2}$

Step 4.3

Simplify $\frac{- 190 \pm 50}{- 2}$ .

$x = 95 \pm 25$

Step 5

The final answer is the combination of both solutions.

$x = 120, 70$