Basic Math Examples

$t + \frac{4}{t - 2} = \frac{1}{4}$

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.

$1, t - 2, 4$

Step 1.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.4

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 1.5

Multiply $2$ by $2$ .

$4$

Step 1.6

The factor for $t - 2$ is $t - 2$ itself.

$(t - 2) = t - 2$

$(t - 2)$ occurs $1$ time.

Step 1.7

The LCM of $t - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$t - 2$

Step 1.8

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$4 (t - 2)$

Step 2

Multiply each term in $t + \frac{4}{t - 2} = \frac{1}{4}$ by $4 (t - 2)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $t + \frac{4}{t - 2} = \frac{1}{4}$ by $4 (t - 2)$ .

$t (4 (t - 2)) + \frac{4}{t - 2} (4 (t - 2)) = \frac{1}{4} (4 (t - 2))$

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

Rewrite using the commutative property of multiplication.

$4 t (t - 2) + \frac{4}{t - 2} (4 (t - 2)) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.2

Apply the distributive property.

$4 t \cdot t + 4 t \cdot - 2 + \frac{4}{t - 2} (4 (t - 2)) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.3

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

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Step 2.2.1.3.1

Move $t$ .

$4 (t \cdot t) + 4 t \cdot - 2 + \frac{4}{t - 2} (4 (t - 2)) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.3.2

Multiply $t$ by $t$ .

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

Step 2.2.1.4

Multiply $- 2$ by $4$ .

$4 t^{2} - 8 t + \frac{4}{t - 2} (4 (t - 2)) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.5

Rewrite using the commutative property of multiplication.

$4 t^{2} - 8 t + 4 \frac{4}{t - 2} (t - 2) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.6

Multiply $4 \frac{4}{t - 2}$ .

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Step 2.2.1.6.1

Combine $4$ and $\frac{4}{t - 2}$ .

$4 t^{2} - 8 t + \frac{4 \cdot 4}{t - 2} (t - 2) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.6.2

Multiply $4$ by $4$ .

$4 t^{2} - 8 t + \frac{16}{t - 2} (t - 2) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.7

Cancel the common factor of $t - 2$ .

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Step 2.2.1.7.1

Cancel the common factor.

$4 t^{2} - 8 t + \frac{16}{t - 2} (t - 2) = \frac{1}{4} (4 (t - 2))$

Step 2.2.1.7.2

Rewrite the expression.

$4 t^{2} - 8 t + 16 = \frac{1}{4} (4 (t - 2))$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $4$ .

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Step 2.3.1.1

Cancel the common factor.

$4 t^{2} - 8 t + 16 = \frac{1}{4} (4 (t - 2))$

Step 2.3.1.2

Rewrite the expression.

$4 t^{2} - 8 t + 16 = t - 2$

Step 3

Solve the equation.

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Step 3.1

Move all terms containing $t$ to the left side of the equation.

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Step 3.1.1

Subtract $t$ from both sides of the equation.

$4 t^{2} - 8 t + 16 - t = - 2$

Step 3.1.2

Subtract $t$ from $- 8 t$ .

$4 t^{2} - 9 t + 16 = - 2$

Step 3.2

Move all terms to the left side of the equation and simplify.

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Step 3.2.1

Add $2$ to both sides of the equation.

$4 t^{2} - 9 t + 16 + 2 = 0$

Step 3.2.2

Add $16$ and $2$ .

$4 t^{2} - 9 t + 18 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = 4$ , $b = - 9$ , and $c = 18$ into the quadratic formula and solve for $t$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (4 \cdot 18)}}{2 \cdot 4}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 9$ to the power of $2$ .

$t = \frac{9 \pm \sqrt{81 - 4 \cdot 4 \cdot 18}}{2 \cdot 4}$

Step 3.5.1.2

Multiply $- 4 \cdot 4 \cdot 18$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{9 \pm \sqrt{81 - 16 \cdot 18}}{2 \cdot 4}$

Step 3.5.1.2.2

Multiply $- 16$ by $18$ .

$t = \frac{9 \pm \sqrt{81 - 288}}{2 \cdot 4}$

Step 3.5.1.3

Subtract $288$ from $81$ .

$t = \frac{9 \pm \sqrt{- 207}}{2 \cdot 4}$

Step 3.5.1.4

Rewrite $- 207$ as $- 1 (207)$ .

$t = \frac{9 \pm \sqrt{- 1 \cdot 207}}{2 \cdot 4}$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (207)}$ as $\sqrt{- 1} \cdot \sqrt{207}$ .

$t = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{207}}{2 \cdot 4}$

Step 3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{9 \pm i \cdot \sqrt{207}}{2 \cdot 4}$

Step 3.5.1.7

Rewrite $207$ as $3^{2} \cdot 23$ .

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Step 3.5.1.7.1

Factor $9$ out of $207$ .

$t = \frac{9 \pm i \cdot \sqrt{9 (23)}}{2 \cdot 4}$

Step 3.5.1.7.2

Rewrite $9$ as $3^{2}$ .

$t = \frac{9 \pm i \cdot \sqrt{3^{2} \cdot 23}}{2 \cdot 4}$

Step 3.5.1.8

Pull terms out from under the radical.

$t = \frac{9 \pm i \cdot (3 \sqrt{23})}{2 \cdot 4}$

Step 3.5.1.9

Move $3$ to the left of $i$ .

$t = \frac{9 \pm 3 i \sqrt{23}}{2 \cdot 4}$

Step 3.5.2

Multiply $2$ by $4$ .

$t = \frac{9 \pm 3 i \sqrt{23}}{8}$

Step 3.6

The final answer is the combination of both solutions.

$t = \frac{9 + 3 i \sqrt{23}}{8}, \frac{9 - 3 i \sqrt{23}}{8}$