Basic Math Examples

$36 t + (\frac{h - g}{t}) = d$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, t, 1$

Step 1.2

The LCM of one and any expression is the expression.

$t$

Step 2

Multiply each term in $36 t + \frac{h - g}{t} = d$ by $t$ to eliminate the fractions.

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Step 2.1

Multiply each term in $36 t + \frac{h - g}{t} = d$ by $t$ .

$36 t \cdot t + \frac{h - g}{t} t = d t$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $t$ by $t$ by adding the exponents.

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Step 2.2.1.1.1

Move $t$ .

$36 (t \cdot t) + \frac{h - g}{t} t = d t$

Step 2.2.1.1.2

Multiply $t$ by $t$ .

$36 t^{2} + \frac{h - g}{t} t = d t$

Step 2.2.1.2

Cancel the common factor of $t$ .

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Step 2.2.1.2.1

Cancel the common factor.

$36 t^{2} + \frac{h - g}{t} t = d t$

Step 2.2.1.2.2

Rewrite the expression.

$36 t^{2} + h - g = d t$

Step 3

Solve the equation.

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Step 3.1

Subtract $d t$ from both sides of the equation.

$36 t^{2} + h - g - d t = 0$

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

Substitute the values $a = 36$ , $b = - d$ , and $c = h - g$ into the quadratic formula and solve for $t$ .

$\frac{d \pm \sqrt{{(- d)}^{2} - 4 \cdot (36 \cdot (h - g))}}{2 \cdot 36}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Apply the product rule to $- d$ .

$t = \frac{d \pm \sqrt{{(- 1)}^{2} d^{2} - 4 \cdot 36 \cdot (h - g)}}{2 \cdot 36}$

Step 3.4.1.2

Raise $- 1$ to the power of $2$ .

$t = \frac{d \pm \sqrt{1 d^{2} - 4 \cdot 36 \cdot (h - g)}}{2 \cdot 36}$

Step 3.4.1.3

Multiply $d^{2}$ by $1$ .

$t = \frac{d \pm \sqrt{d^{2} - 4 \cdot 36 \cdot (h - g)}}{2 \cdot 36}$

Step 3.4.1.4

Multiply $- 4$ by $36$ .

$t = \frac{d \pm \sqrt{d^{2} - 144 \cdot (h - g)}}{2 \cdot 36}$

Step 3.4.1.5

Apply the distributive property.

$t = \frac{d \pm \sqrt{d^{2} - 144 h - 144 (- g)}}{2 \cdot 36}$

Step 3.4.1.6

Multiply $- 1$ by $- 144$ .

$t = \frac{d \pm \sqrt{d^{2} - 144 h + 144 g}}{2 \cdot 36}$

Step 3.4.2

Multiply $2$ by $36$ .

$t = \frac{d \pm \sqrt{d^{2} - 144 h + 144 g}}{72}$

Step 3.5

The final answer is the combination of both solutions.

$t = \frac{d + \sqrt{d^{2} - 144 h + 144 g}}{72}$

$t = \frac{d - \sqrt{d^{2} - 144 h + 144 g}}{72}$

$t = \frac{d + \sqrt{d^{2} - 144 h + 144 g}}{72}$

$t = \frac{d - \sqrt{d^{2} - 144 h + 144 g}}{72}$