Finite Math Examples

$53 = \frac{170 t}{t^{2} + 1}$

Step 1

Rewrite the equation as $\frac{170 t}{t^{2} + 1} = 53$ .

$\frac{170 t}{t^{2} + 1} = 53$

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.

$t^{2} + 1, 1$

Step 2.2

Remove parentheses.

$t^{2} + 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$t^{2} + 1$

Step 3

Multiply each term in $\frac{170 t}{t^{2} + 1} = 53$ by $t^{2} + 1$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{170 t}{t^{2} + 1} = 53$ by $t^{2} + 1$ .

$\frac{170 t}{t^{2} + 1} (t^{2} + 1) = 53 (t^{2} + 1)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $t^{2} + 1$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{170 t}{t^{2} + 1} (t^{2} + 1) = 53 (t^{2} + 1)$

Step 3.2.1.2

Rewrite the expression.

$170 t = 53 (t^{2} + 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$170 t = 53 t^{2} + 53 \cdot 1$

Step 3.3.2

Multiply $53$ by $1$ .

$170 t = 53 t^{2} + 53$

Step 4

Solve the equation.

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Step 4.1

Subtract $53 t^{2}$ from both sides of the equation.

$170 t - 53 t^{2} = 53$

Step 4.2

Subtract $53$ from both sides of the equation.

$170 t - 53 t^{2} - 53 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = - 53$ , $b = 170$ , and $c = - 53$ into the quadratic formula and solve for $t$ .

$\frac{- 170 \pm \sqrt{170^{2} - 4 \cdot (- 53 \cdot - 53)}}{2 \cdot - 53}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $170$ to the power of $2$ .

$t = \frac{- 170 \pm \sqrt{28900 - 4 \cdot - 53 \cdot - 53}}{2 \cdot - 53}$

Step 4.5.1.2

Multiply $- 4 \cdot - 53 \cdot - 53$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $- 53$ .

$t = \frac{- 170 \pm \sqrt{28900 + 212 \cdot - 53}}{2 \cdot - 53}$

Step 4.5.1.2.2

Multiply $212$ by $- 53$ .

$t = \frac{- 170 \pm \sqrt{28900 - 11236}}{2 \cdot - 53}$

Step 4.5.1.3

Subtract $11236$ from $28900$ .

$t = \frac{- 170 \pm \sqrt{17664}}{2 \cdot - 53}$

Step 4.5.1.4

Rewrite $17664$ as $16^{2} \cdot 69$ .

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Step 4.5.1.4.1

Factor $256$ out of $17664$ .

$t = \frac{- 170 \pm \sqrt{256 (69)}}{2 \cdot - 53}$

Step 4.5.1.4.2

Rewrite $256$ as $16^{2}$ .

$t = \frac{- 170 \pm \sqrt{16^{2} \cdot 69}}{2 \cdot - 53}$

Step 4.5.1.5

Pull terms out from under the radical.

$t = \frac{- 170 \pm 16 \sqrt{69}}{2 \cdot - 53}$

Step 4.5.2

Multiply $2$ by $- 53$ .

$t = \frac{- 170 \pm 16 \sqrt{69}}{- 106}$

Step 4.5.3

Simplify $\frac{- 170 \pm 16 \sqrt{69}}{- 106}$ .

$t = \frac{85 \pm 8 \sqrt{69}}{53}$

Step 4.6

The final answer is the combination of both solutions.

$t = \frac{85 + 8 \sqrt{69}}{53}, \frac{85 - 8 \sqrt{69}}{53}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{85 + 8 \sqrt{69}}{53}, \frac{85 - 8 \sqrt{69}}{53}$

Decimal Form:

$t = 2.85760360 \dots, 0.34994356 \dots$