Algebra Examples

$s = 2 p r h + p r^{2}$

Step 1

Rewrite the equation as $2 p r h + p r^{2} = s$ .

$2 p r h + p r^{2} = s$

Step 2

Subtract $s$ from both sides of the equation.

$2 p r h + p r^{2} - s = 0$

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 $a = p$ , $b = 2 p h$ , and $c = - s$ into the quadratic formula and solve for $r$ .

$\frac{- 2 p h \pm \sqrt{{(2 p h)}^{2} - 4 \cdot (p \cdot (- s))}}{2 p}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add parentheses.

$r = \frac{- 2 p h \pm \sqrt{{(2 (p h))}^{2} - 4 \cdot (p \cdot (- s))}}{2 p}$

Step 5.1.2

Let $u = p \cdot (- s)$ . Substitute $u$ for all occurrences of $p \cdot (- s)$ .

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Step 5.1.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.2.1.1

Apply the product rule to $2 p h$ .

$r = \frac{- 2 p h \pm \sqrt{{(2 p)}^{2} h^{2} - 4 \cdot u}}{2 p}$

Step 5.1.2.1.2

Apply the product rule to $2 p$ .

$r = \frac{- 2 p h \pm \sqrt{2^{2} p^{2} h^{2} - 4 \cdot u}}{2 p}$

Step 5.1.2.2

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

$r = \frac{- 2 p h \pm \sqrt{4 p^{2} h^{2} - 4 u}}{2 p}$

Step 5.1.3

Factor $4$ out of $4 p^{2} h^{2} - 4 u$ .

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Step 5.1.3.1

Factor $4$ out of $4 p^{2} h^{2}$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2}) - 4 u}}{2 p}$

Step 5.1.3.2

Factor $4$ out of $- 4 u$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2}) + 4 (- u)}}{2 p}$

Step 5.1.3.3

Factor $4$ out of $4 (p^{2} h^{2}) + 4 (- u)$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2} - u)}}{2 p}$

Step 5.1.4

Replace all occurrences of $u$ with $p \cdot (- s)$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2} - (p \cdot (- s)))}}{2 p}$

Step 5.1.5

Simplify each term.

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Step 5.1.5.1

Rewrite using the commutative property of multiplication.

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2} - (- p s))}}{2 p}$

Step 5.1.5.2

Multiply $- (- p s)$ .

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Step 5.1.5.2.1

Multiply $- 1$ by $- 1$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2} + 1 (p s))}}{2 p}$

Step 5.1.5.2.2

Multiply $p$ by $1$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p^{2} h^{2} + p s)}}{2 p}$

Step 5.1.6

Factor $p$ out of $p^{2} h^{2} + p s$ .

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Step 5.1.6.1

Factor $p$ out of $p^{2} h^{2}$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p (p h^{2}) + p s)}}{2 p}$

Step 5.1.6.2

Factor $p$ out of $p s$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p (p h^{2}) + p (s))}}{2 p}$

Step 5.1.6.3

Factor $p$ out of $p (p h^{2}) + p (s)$ .

$r = \frac{- 2 p h \pm \sqrt{4 (p (p h^{2} + s))}}{2 p}$

$r = \frac{- 2 p h \pm \sqrt{4 p (p h^{2} + s)}}{2 p}$

Step 5.1.7

Rewrite $4 p (p h^{2} + s)$ as $2^{2} (p (p h^{2} + s))$ .

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Step 5.1.7.1

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

$r = \frac{- 2 p h \pm \sqrt{2^{2} p (p h^{2} + s)}}{2 p}$

Step 5.1.7.2

Add parentheses.

$r = \frac{- 2 p h \pm \sqrt{2^{2} (p (p h^{2} + s))}}{2 p}$

Step 5.1.8

Pull terms out from under the radical.

$r = \frac{- 2 p h \pm 2 \sqrt{p (p h^{2} + s)}}{2 p}$

Step 5.2

Simplify $\frac{- 2 p h \pm 2 \sqrt{p (p h^{2} + s)}}{2 p}$ .

$r = \frac{- p h \pm \sqrt{p (p h^{2} + s)}}{p}$

Step 6

The final answer is the combination of both solutions.

$r = - \frac{p h - \sqrt{p (p h^{2} + s)}}{p}$

$r = - \frac{p h + \sqrt{p (p h^{2} + s)}}{p}$