Basic Math Examples

$A = \frac{1}{2} \cdot ((b + b^{2}) h)$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot ((b + b^{2}) h) = A$ .

$\frac{1}{2} \cdot ((b + b^{2}) h) = A$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot ((b + b^{2}) h)) = 2 A$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 (\frac{1}{2} \cdot ((b + b^{2}) h))$ .

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Step 3.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (b h + b^{2} h)) = 2 A$

Step 3.1.2

Apply the distributive property.

$2 (\frac{1}{2} (b h) + \frac{1}{2} (b^{2} h)) = 2 A$

Step 3.1.3

Multiply $\frac{1}{2} (b h)$ .

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Step 3.1.3.1

Combine $b$ and $\frac{1}{2}$ .

$2 (\frac{b}{2} h + \frac{1}{2} (b^{2} h)) = 2 A$

Step 3.1.3.2

Combine $\frac{b}{2}$ and $h$ .

$2 (\frac{b h}{2} + \frac{1}{2} (b^{2} h)) = 2 A$

Step 3.1.4

Multiply $\frac{1}{2} (b^{2} h)$ .

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Step 3.1.4.1

Combine $b^{2}$ and $\frac{1}{2}$ .

$2 (\frac{b h}{2} + \frac{b^{2}}{2} h) = 2 A$

Step 3.1.4.2

Combine $\frac{b^{2}}{2}$ and $h$ .

$2 (\frac{b h}{2} + \frac{b^{2} h}{2}) = 2 A$

Step 3.1.5

Apply the distributive property.

$2 \frac{b h}{2} + 2 \frac{b^{2} h}{2} = 2 A$

Step 3.1.6

Cancel the common factor of $2$ .

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Step 3.1.6.1

Cancel the common factor.

$2 \frac{b h}{2} + 2 \frac{b^{2} h}{2} = 2 A$

Step 3.1.6.2

Rewrite the expression.

$b h + 2 \frac{b^{2} h}{2} = 2 A$

Step 3.1.7

Cancel the common factor of $2$ .

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Step 3.1.7.1

Cancel the common factor.

$b h + 2 \frac{b^{2} h}{2} = 2 A$

Step 3.1.7.2

Rewrite the expression.

$b h + b^{2} h = 2 A$

Step 4

Subtract $2 A$ from both sides of the equation.

$b h + b^{2} h - 2 A = 0$

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

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

Step 7

Simplify the numerator.

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Step 7.1

Factor $h$ out of $h^{2} - 4 \cdot h \cdot - 2 A$ .

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Step 7.1.1

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

$b = \frac{- h \pm \sqrt{h \cdot h - 4 \cdot h \cdot (- 2 A)}}{2 h}$

Step 7.1.2

Factor $h$ out of $- 4 \cdot h \cdot - 2 A$ .

$b = \frac{- h \pm \sqrt{h \cdot h + h (- 4 \cdot (- 2 A))}}{2 h}$

Step 7.1.3

Factor $h$ out of $h \cdot h + h (- 4 \cdot - 2 A)$ .

$b = \frac{- h \pm \sqrt{h (h - 4 \cdot (- 2 A))}}{2 h}$

Step 7.2

Multiply $- 4$ by $- 2$ .

$b = \frac{- h \pm \sqrt{h (h + 8 A)}}{2 h}$

Step 8

The final answer is the combination of both solutions.

$b = - \frac{h - \sqrt{h (h + 8 A)}}{2 h}$

$b = - \frac{h + \sqrt{h (h + 8 A)}}{2 h}$