Basic Math Examples

$5 k = 3 t - \frac{3 k}{t}$

Step 1

Rewrite the equation as $3 t - \frac{3 k}{t} = 5 k$ .

$3 t - \frac{3 k}{t} = 5 k$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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 2.2

The LCM of one and any expression is the expression.

$t$

Step 3

Multiply each term in $3 t - \frac{3 k}{t} = 5 k$ by $t$ to eliminate the fractions.

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Step 3.1

Multiply each term in $3 t - \frac{3 k}{t} = 5 k$ by $t$ .

$3 t \cdot t - \frac{3 k}{t} t = 5 k t$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Move $t$ .

$3 (t \cdot t) - \frac{3 k}{t} t = 5 k t$

Step 3.2.1.1.2

Multiply $t$ by $t$ .

$3 t^{2} - \frac{3 k}{t} t = 5 k t$

Step 3.2.1.2

Cancel the common factor of $t$ .

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Step 3.2.1.2.1

Move the leading negative in $- \frac{3 k}{t}$ into the numerator.

$3 t^{2} + \frac{- 3 k}{t} t = 5 k t$

Step 3.2.1.2.2

Cancel the common factor.

$3 t^{2} + \frac{- 3 k}{t} t = 5 k t$

Step 3.2.1.2.3

Rewrite the expression.

$3 t^{2} - 3 k = 5 k t$

Step 4

Solve the equation.

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Step 4.1

Subtract $5 k t$ from both sides of the equation.

$3 t^{2} - 3 k - 5 k t = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = 3$ , $b = - 5 k$ , and $c = - 3 k$ into the quadratic formula and solve for $t$ .

$\frac{5 k \pm \sqrt{{(- 5 k)}^{2} - 4 \cdot (3 \cdot (- 3 k))}}{2 \cdot 3}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Apply the product rule to $- 5 k$ .

$t = \frac{5 k \pm \sqrt{{(- 5)}^{2} k^{2} - 4 \cdot 3 \cdot (- 3 k)}}{2 \cdot 3}$

Step 4.4.1.2

Raise $- 5$ to the power of $2$ .

$t = \frac{5 k \pm \sqrt{25 k^{2} - 4 \cdot 3 \cdot (- 3 k)}}{2 \cdot 3}$

Step 4.4.1.3

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

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Step 4.4.1.3.1

Multiply $- 4$ by $3$ .

$t = \frac{5 k \pm \sqrt{25 k^{2} - 12 \cdot (- 3 k)}}{2 \cdot 3}$

Step 4.4.1.3.2

Multiply $- 12$ by $- 3$ .

$t = \frac{5 k \pm \sqrt{25 k^{2} + 36 k}}{2 \cdot 3}$

Step 4.4.1.4

Factor $k$ out of $25 k^{2} + 36 k$ .

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Step 4.4.1.4.1

Factor $k$ out of $25 k^{2}$ .

$t = \frac{5 k \pm \sqrt{k (25 k) + 36 k}}{2 \cdot 3}$

Step 4.4.1.4.2

Factor $k$ out of $36 k$ .

$t = \frac{5 k \pm \sqrt{k (25 k) + k \cdot 36}}{2 \cdot 3}$

Step 4.4.1.4.3

Factor $k$ out of $k (25 k) + k \cdot 36$ .

$t = \frac{5 k \pm \sqrt{k (25 k + 36)}}{2 \cdot 3}$

Step 4.4.2

Multiply $2$ by $3$ .

$t = \frac{5 k \pm \sqrt{k (25 k + 36)}}{6}$

Step 4.5

The final answer is the combination of both solutions.

$t = \frac{5 k + \sqrt{k (25 k + 36)}}{6}$

$t = \frac{5 k - \sqrt{k (25 k + 36)}}{6}$

$t = \frac{5 k + \sqrt{k (25 k + 36)}}{6}$

$t = \frac{5 k - \sqrt{k (25 k + 36)}}{6}$