Finite Math Examples

$R (3) = x (60 - 3 x)$

Step 1

Rewrite the equation as $x (60 - 3 x) = R (3)$ .

$x (60 - 3 x) = R (3)$

Step 2

Simplify $x (60 - 3 x)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$x \cdot 60 + x (- 3 x) = R (3)$

Step 2.1.2

Reorder.

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Step 2.1.2.1

Move $60$ to the left of $x$ .

$60 \cdot x + x (- 3 x) = R (3)$

Step 2.1.2.2

Rewrite using the commutative property of multiplication.

$60 \cdot x - 3 x \cdot x = R (3)$

Step 2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1

Move $x$ .

$60 x - 3 (x \cdot x) = R (3)$

Step 2.2.2

Multiply $x$ by $x$ .

$60 x - 3 x^{2} = R (3)$

Step 3

Move $3$ to the left of $R$ .

$60 x - 3 x^{2} = 3 R$

Step 4

Subtract $3 R$ from both sides of the equation.

$60 x - 3 x^{2} - 3 R = 0$

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 $a = - 3$ , $b = 60$ , and $c = - 3 R$ into the quadratic formula and solve for $x$ .

$\frac{- 60 \pm \sqrt{60^{2} - 4 \cdot (- 3 \cdot (- 3 R))}}{2 \cdot - 3}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $60$ to the power of $2$ .

$x = \frac{- 60 \pm \sqrt{3600 - 4 \cdot - 3 \cdot (- 3 R)}}{2 \cdot - 3}$

Step 7.1.2

Multiply $- 4 \cdot - 3 \cdot - 3$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 60 \pm \sqrt{3600 + 12 \cdot (- 3 R)}}{2 \cdot - 3}$

Step 7.1.2.2

Multiply $12$ by $- 3$ .

$x = \frac{- 60 \pm \sqrt{3600 - 36 R}}{2 \cdot - 3}$

Step 7.1.3

Factor $36$ out of $3600 - 36 R$ .

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Step 7.1.3.1

Factor $36$ out of $3600$ .

$x = \frac{- 60 \pm \sqrt{36 (100) - 36 R}}{2 \cdot - 3}$

Step 7.1.3.2

Factor $36$ out of $- 36 R$ .

$x = \frac{- 60 \pm \sqrt{36 (100) + 36 (- R)}}{2 \cdot - 3}$

Step 7.1.3.3

Factor $36$ out of $36 (100) + 36 (- R)$ .

$x = \frac{- 60 \pm \sqrt{36 (100 - R)}}{2 \cdot - 3}$

Step 7.1.4

Rewrite $36 (100 - R)$ as $6^{2} (10^{2} - R)$ .

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Step 7.1.4.1

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 60 \pm \sqrt{6^{2} (100 - R)}}{2 \cdot - 3}$

Step 7.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 60 \pm \sqrt{6^{2} (10^{2} - R)}}{2 \cdot - 3}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 60 \pm 6 \sqrt{10^{2} - R}}{2 \cdot - 3}$

Step 7.1.6

Raise $10$ to the power of $2$ .

$x = \frac{- 60 \pm 6 \sqrt{100 - R}}{2 \cdot - 3}$

Step 7.2

Multiply $2$ by $- 3$ .

$x = \frac{- 60 \pm 6 \sqrt{100 - R}}{- 6}$

Step 7.3

Simplify $\frac{- 60 \pm 6 \sqrt{100 - R}}{- 6}$ .

$x = 10 \pm \sqrt{100 - R}$

Step 8

The final answer is the combination of both solutions.

$x = 10 + \sqrt{100 - R}$

$x = 10 - \sqrt{100 - R}$