Basic Math Examples

$\frac{4 t^{2}}{5} = \frac{9 t}{5} + \frac{11}{10}$

Step 1

Multiply both sides by $5$ .

$\frac{4 t^{2}}{5} \cdot 5 = (\frac{9 t}{5} + \frac{11}{10}) \cdot 5$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $5$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{4 t^{2}}{5} \cdot 5 = (\frac{9 t}{5} + \frac{11}{10}) \cdot 5$

Step 2.1.1.2

Rewrite the expression.

$4 t^{2} = (\frac{9 t}{5} + \frac{11}{10}) \cdot 5$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{9 t}{5} + \frac{11}{10}) \cdot 5$ .

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Step 2.2.1.1

Apply the distributive property.

$4 t^{2} = \frac{9 t}{5} \cdot 5 + \frac{11}{10} \cdot 5$

Step 2.2.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.2.1

Cancel the common factor.

$4 t^{2} = \frac{9 t}{5} \cdot 5 + \frac{11}{10} \cdot 5$

Step 2.2.1.2.2

Rewrite the expression.

$4 t^{2} = 9 t + \frac{11}{10} \cdot 5$

Step 2.2.1.3

Cancel the common factor of $5$ .

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Step 2.2.1.3.1

Factor $5$ out of $10$ .

$4 t^{2} = 9 t + \frac{11}{5 (2)} \cdot 5$

Step 2.2.1.3.2

Cancel the common factor.

$4 t^{2} = 9 t + \frac{11}{5 \cdot 2} \cdot 5$

Step 2.2.1.3.3

Rewrite the expression.

$4 t^{2} = 9 t + \frac{11}{2}$

Step 3

Solve for $t$ .

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Step 3.1

Subtract $9 t$ from both sides of the equation.

$4 t^{2} - 9 t = \frac{11}{2}$

Step 3.2

Subtract $\frac{11}{2}$ from both sides of the equation.

$4 t^{2} - 9 t - \frac{11}{2} = 0$

Step 3.3

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.3.1

Apply the distributive property.

$2 (4 t^{2}) + 2 (- 9 t) + 2 (- \frac{11}{2}) = 0$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $4$ by $2$ .

$8 t^{2} + 2 (- 9 t) + 2 (- \frac{11}{2}) = 0$

Step 3.3.2.2

Multiply $- 9$ by $2$ .

$8 t^{2} - 18 t + 2 (- \frac{11}{2}) = 0$

Step 3.3.2.3

Cancel the common factor of $2$ .

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Step 3.3.2.3.1

Move the leading negative in $- \frac{11}{2}$ into the numerator.

$8 t^{2} - 18 t + 2 (\frac{- 11}{2}) = 0$

Step 3.3.2.3.2

Cancel the common factor.

$8 t^{2} - 18 t + 2 (\frac{- 11}{2}) = 0$

Step 3.3.2.3.3

Rewrite the expression.

$8 t^{2} - 18 t - 11 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 8$ , $b = - 18$ , and $c = - 11$ into the quadratic formula and solve for $t$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (8 \cdot - 11)}}{2 \cdot 8}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 18$ to the power of $2$ .

$t = \frac{18 \pm \sqrt{324 - 4 \cdot 8 \cdot - 11}}{2 \cdot 8}$

Step 3.6.1.2

Multiply $- 4 \cdot 8 \cdot - 11$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $8$ .

$t = \frac{18 \pm \sqrt{324 - 32 \cdot - 11}}{2 \cdot 8}$

Step 3.6.1.2.2

Multiply $- 32$ by $- 11$ .

$t = \frac{18 \pm \sqrt{324 + 352}}{2 \cdot 8}$

Step 3.6.1.3

Add $324$ and $352$ .

$t = \frac{18 \pm \sqrt{676}}{2 \cdot 8}$

Step 3.6.1.4

Rewrite $676$ as $26^{2}$ .

$t = \frac{18 \pm \sqrt{26^{2}}}{2 \cdot 8}$

Step 3.6.1.5

Pull terms out from under the radical, assuming positive real numbers.

$t = \frac{18 \pm 26}{2 \cdot 8}$

Step 3.6.2

Multiply $2$ by $8$ .

$t = \frac{18 \pm 26}{16}$

Step 3.6.3

Simplify $\frac{18 \pm 26}{16}$ .

$t = \frac{9 \pm 13}{8}$

Step 3.7

The final answer is the combination of both solutions.

$t = \frac{11}{4}, - \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{11}{4}, - \frac{1}{2}$

Decimal Form:

$t = 2.75, - 0.5$

Mixed Number Form:

$t = 2 \frac{3}{4}, - \frac{1}{2}$