Solve for t 0.5=(6t)/(t^2+4)

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

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

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

Solve for $t$ .

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Multiply each term by $t^{2} + 4$ and simplify.

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

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

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

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify $0.5 \cdot (t^{2} + 4)$ .

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Apply the distributive property.

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

Multiply $0.5$ by $4$ .

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

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

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

Move $2$ to the left side of the equation by subtracting it from both sides.

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

Factor the left side of the equation.

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

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Factor $0.5$ out of $6 t$ .

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

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

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

Factor $0.5$ out of $- 2$ .

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

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

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

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

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

Reorder terms.

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

Replace the left side with the factored expression.

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

Divide each term by $0.5$ and simplify.

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

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

Cancel the common factor of $0.5$ .

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

Divide $0$ by $0.5$ .

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

Factor $- 1$ out of $- t^{2} + 12 t - 4$ .

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Factor $- 1$ out of $- t^{2}$ .

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

Factor $- 1$ out of $12 t$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

$- (t^{2}) - (- 12 t) - 1 \cdot 4 = 0$

Factor $- 1$ out of $- (t^{2}) - (- 12 t)$ .

$- (t^{2} - 12 t) - 1 \cdot 4 = 0$

Factor $- 1$ out of $- (t^{2} - 12 t) - 1 (4)$ .

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

Multiply each term in $- (t^{2} - 12 t + 4) = 0$ by $- 1$

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Multiply each term in $- (t^{2} - 12 t + 4) = 0$ by $- 1$ .

$- (t^{2} - 12 t + 4) \cdot - 1 = 0 \cdot - 1$

Simplify $- (t^{2} - 12 t + 4) \cdot - 1$ .

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Apply the distributive property.

$(- t^{2} - (- 12 t) - 1 \cdot 4) \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $- 12$ by $- 1$ .

$(- t^{2} + 12 t - 1 \cdot 4) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $4$ .

$(- t^{2} + 12 t - 4) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$- t^{2} \cdot - 1 + 12 t \cdot - 1 - 4 \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $- t^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 t^{2} + 12 t \cdot - 1 - 4 \cdot - 1 = 0 \cdot - 1$

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

$t^{2} + 12 t \cdot - 1 - 4 \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $12$ .

$t^{2} - 12 t - 4 \cdot - 1 = 0 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$t^{2} - 12 t + 4 = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

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Simplify the numerator.

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

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

Multiply $4$ by $1$ .

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

Multiply $- 4$ by $4$ .

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

Subtract $16$ from $144$ .

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

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

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Factor $64$ out of $128$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

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

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

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

The final answer is the combination of both solutions.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$t = 11.65685424 \dots, 0.34314575 \dots$

Solve for t 0.5=(6t)/(t^2+4)

Solving MATH problems