Basic Math Examples

$0.5 = \frac{6 t}{t^{2} + 4}$

Step 1

Rewrite the equation as $\frac{6 t}{t^{2} + 4} = 0.5$ .

$\frac{6 t}{t^{2} + 4} = 0.5$

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.

$t^{2} + 4, 1$

Step 2.2

Remove parentheses.

$t^{2} + 4, 1$

Step 2.3

The LCM of one and any expression is the expression.

$t^{2} + 4$

Step 3

Multiply each term in $\frac{6 t}{t^{2} + 4} = 0.5$ by $t^{2} + 4$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{6 t}{t^{2} + 4} = 0.5$ by $t^{2} + 4$ .

$\frac{6 t}{t^{2} + 4} (t^{2} + 4) = 0.5 (t^{2} + 4)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $t^{2} + 4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 t}{t^{2} + 4} (t^{2} + 4) = 0.5 (t^{2} + 4)$

Step 3.2.1.2

Rewrite the expression.

$6 t = 0.5 (t^{2} + 4)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$6 t = 0.5 t^{2} + 0.5 \cdot 4$

Step 3.3.2

Multiply $0.5$ by $4$ .

$6 t = 0.5 t^{2} + 2$

Step 4

Solve the equation.

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Step 4.1

Subtract $0.5 t^{2}$ from both sides of the equation.

$6 t - 0.5 t^{2} = 2$

Step 4.2

Subtract $2$ from both sides of the equation.

$6 t - 0.5 t^{2} - 2 = 0$

Step 4.3

Factor $- 0.5$ out of $6 t - 0.5 t^{2} - 2$ .

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Step 4.3.1

Reorder $6 t$ and $- 0.5 t^{2}$ .

$- 0.5 t^{2} + 6 t - 2 = 0$

Step 4.3.2

Factor $- 0.5$ out of $- 0.5 t^{2}$ .

$- 0.5 t^{2} + 6 t - 2 = 0$

Step 4.3.3

Factor $- 0.5$ out of $6 t$ .

$- 0.5 t^{2} - 0.5 (- 12 t) - 2 = 0$

Step 4.3.4

Factor $- 0.5$ out of $- 2$ .

$- 0.5 t^{2} - 0.5 (- 12 t) - 0.5 \cdot 4 = 0$

Step 4.3.5

Factor $- 0.5$ out of $- 0.5 (t^{2}) - 0.5 (- 12 t)$ .

$- 0.5 (t^{2} - 12 t) - 0.5 \cdot 4 = 0$

Step 4.3.6

Factor $- 0.5$ out of $- 0.5 (t^{2} - 12 t) - 0.5 (4)$ .

$- 0.5 (t^{2} - 12 t + 4) = 0$

Step 4.4

Divide each term in $- 0.5 (t^{2} - 12 t + 4) = 0$ by $- 0.5$ and simplify.

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Step 4.4.1

Divide each term in $- 0.5 (t^{2} - 12 t + 4) = 0$ by $- 0.5$ .

$\frac{- 0.5 (t^{2} - 12 t + 4)}{- 0.5} = \frac{0}{- 0.5}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $- 0.5$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{- 0.5 (t^{2} - 12 t + 4)}{- 0.5} = \frac{0}{- 0.5}$

Step 4.4.2.1.2

Divide $t^{2} - 12 t + 4$ by $1$ .

$t^{2} - 12 t + 4 = \frac{0}{- 0.5}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $- 0.5$ .

$t^{2} - 12 t + 4 = 0$

Step 4.5

Use the quadratic formula to find the solutions.

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

Step 4.6

Substitute the values $a = 1$ , $b = - 12$ , and $c = 4$ into the quadratic formula and solve for $t$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 4.7

Simplify.

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Raise $- 12$ to the power of $2$ .

$t = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 4.7.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{12 \pm \sqrt{144 - 4 \cdot 4}}{2 \cdot 1}$

Step 4.7.1.2.2

Multiply $- 4$ by $4$ .

$t = \frac{12 \pm \sqrt{144 - 16}}{2 \cdot 1}$

Step 4.7.1.3

Subtract $16$ from $144$ .

$t = \frac{12 \pm \sqrt{128}}{2 \cdot 1}$

Step 4.7.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 4.7.1.4.1

Factor $64$ out of $128$ .

$t = \frac{12 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Step 4.7.1.4.2

Rewrite $64$ as $8^{2}$ .

$t = \frac{12 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Step 4.7.1.5

Pull terms out from under the radical.

$t = \frac{12 \pm 8 \sqrt{2}}{2 \cdot 1}$

Step 4.7.2

Multiply $2$ by $1$ .

$t = \frac{12 \pm 8 \sqrt{2}}{2}$

Step 4.7.3

Simplify $\frac{12 \pm 8 \sqrt{2}}{2}$ .

$t = 6 \pm 4 \sqrt{2}$

Step 4.8

The final answer is the combination of both solutions.

$t = 6 + 4 \sqrt{2}, 6 - 4 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = 6 + 4 \sqrt{2}, 6 - 4 \sqrt{2}$

Decimal Form:

$t = 11.65685424 \dots, 0.34314575 \dots$