Basic Math Examples

$c = 10 + 15 Q - 5 Q^{2} + \frac{Q s}{3}$

Step 1

Rewrite the equation as $10 + 15 Q - 5 Q^{2} + \frac{Q s}{3} = c$ .

$10 + 15 Q - 5 Q^{2} + \frac{Q s}{3} = c$

Step 2

Subtract $c$ from both sides of the equation.

$10 + 15 Q - 5 Q^{2} + \frac{Q s}{3} - c = 0$

Step 3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.1

Apply the distributive property.

$3 \cdot 10 + 3 (15 Q) + 3 (- 5 Q^{2}) + 3 (\frac{Q s}{3}) + 3 (- c) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $3$ by $10$ .

$30 + 3 (15 Q) + 3 (- 5 Q^{2}) + 3 (\frac{Q s}{3}) + 3 (- c) = 0$

Step 3.2.2

Multiply $15$ by $3$ .

$30 + 45 Q + 3 (- 5 Q^{2}) + 3 (\frac{Q s}{3}) + 3 (- c) = 0$

Step 3.2.3

Multiply $- 5$ by $3$ .

$30 + 45 Q - 15 Q^{2} + 3 (\frac{Q s}{3}) + 3 (- c) = 0$

Step 3.2.4

Cancel the common factor of $3$ .

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Step 3.2.4.1

Cancel the common factor.

$30 + 45 Q - 15 Q^{2} + 3 (\frac{Q s}{3}) + 3 (- c) = 0$

Step 3.2.4.2

Rewrite the expression.

$30 + 45 Q - 15 Q^{2} + Q s + 3 (- c) = 0$

Step 3.2.5

Multiply $- 1$ by $3$ .

$30 + 45 Q - 15 Q^{2} + Q s - 3 c = 0$

Step 3.3

Reorder $Q$ and $s$ .

$30 + 45 Q - 15 Q^{2} + s Q - 3 c = 0$

Step 3.4

Move $30$ .

$45 Q - 15 Q^{2} + s Q - 3 c + 30 = 0$

Step 3.5

Move $45 Q$ .

$- 15 Q^{2} + s Q - 3 c + 45 Q + 30 = 0$

Step 3.6

Reorder $- 15 Q^{2}$ and $s Q$ .

$s Q - 15 Q^{2} - 3 c + 45 Q + 30 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 15$ , $b = s + 45$ , and $c = - 3 c + 30$ into the quadratic formula and solve for $Q$ .

$\frac{- (s + 45) \pm \sqrt{{(s + 45)}^{2} - 4 \cdot (- 15 \cdot (- 3 c + 30))}}{2 \cdot - 15}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$Q = \frac{- s - 1 \cdot 45 \pm \sqrt{{(s + 45)}^{2} - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.2

Multiply $- 1$ by $45$ .

$Q = \frac{- s - 45 \pm \sqrt{{(s + 45)}^{2} - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.3

Rewrite ${(s + 45)}^{2}$ as $(s + 45) (s + 45)$ .

$Q = \frac{- s - 45 \pm \sqrt{(s + 45) (s + 45) - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.4

Expand $(s + 45) (s + 45)$ using the FOIL Method.

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Step 6.1.4.1

Apply the distributive property.

$Q = \frac{- s - 45 \pm \sqrt{s (s + 45) + 45 (s + 45) - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.4.2

Apply the distributive property.

$Q = \frac{- s - 45 \pm \sqrt{s \cdot s + s \cdot 45 + 45 (s + 45) - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.4.3

Apply the distributive property.

$Q = \frac{- s - 45 \pm \sqrt{s \cdot s + s \cdot 45 + 45 s + 45 \cdot 45 - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.5

Simplify and combine like terms.

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Step 6.1.5.1

Simplify each term.

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Step 6.1.5.1.1

Multiply $s$ by $s$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + s \cdot 45 + 45 s + 45 \cdot 45 - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.5.1.2

Move $45$ to the left of $s$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 45 \cdot s + 45 s + 45 \cdot 45 - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.5.1.3

Multiply $45$ by $45$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 45 s + 45 s + 2025 - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.5.2

Add $45 s$ and $45 s$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s + 2025 - 4 \cdot - 15 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.6

Multiply $- 4$ by $- 15$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s + 2025 + 60 \cdot (- 3 c + 30)}}{2 \cdot - 15}$

Step 6.1.7

Apply the distributive property.

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s + 2025 + 60 (- 3 c) + 60 \cdot 30}}{2 \cdot - 15}$

Step 6.1.8

Multiply $- 3$ by $60$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s + 2025 - 180 c + 60 \cdot 30}}{2 \cdot - 15}$

Step 6.1.9

Multiply $60$ by $30$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s + 2025 - 180 c + 1800}}{2 \cdot - 15}$

Step 6.1.10

Add $2025$ and $1800$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s - 180 c + 3825}}{2 \cdot - 15}$

Step 6.2

Multiply $2$ by $- 15$ .

$Q = \frac{- s - 45 \pm \sqrt{s^{2} + 90 s - 180 c + 3825}}{- 30}$

Step 6.3

Simplify $\frac{- s - 45 \pm \sqrt{s^{2} + 90 s - 180 c + 3825}}{- 30}$ .

$Q = \frac{s + 45 \pm \sqrt{s^{2} + 90 s - 180 c + 3825}}{30}$

Step 7

The final answer is the combination of both solutions.

$Q = \frac{s + 45 + \sqrt{s^{2} + 90 s - 180 c + 3825}}{30}$

$Q = \frac{s + 45 - \sqrt{s^{2} + 90 s - 180 c + 3825}}{30}$