Solve for t (4t^2)/5=(11t)/5+13/10

$\frac{4 t^{2}}{5} = \frac{11 t}{5} + \frac{13}{10}$

Step 1

Multiply both sides by $5$ .

$\frac{4 t^{2}}{5} \cdot 5 = (\frac{11 t}{5} + \frac{13}{10}) \cdot 5$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{4 t^{2}}{5} \cdot 5 = (\frac{11 t}{5} + \frac{13}{10}) \cdot 5$

Rewrite the expression.

$4 t^{2} = (\frac{11 t}{5} + \frac{13}{10}) \cdot 5$

Simplify the right side.

Simplify $(\frac{11 t}{5} + \frac{13}{10}) \cdot 5$ .

Apply the distributive property.

$4 t^{2} = \frac{11 t}{5} \cdot 5 + \frac{13}{10} \cdot 5$

Cancel the common factor of $5$ .

Cancel the common factor.

$4 t^{2} = \frac{11 t}{5} \cdot 5 + \frac{13}{10} \cdot 5$

Rewrite the expression.

$4 t^{2} = 11 t + \frac{13}{10} \cdot 5$

Cancel the common factor of $5$ .

Factor $5$ out of $10$ .

$4 t^{2} = 11 t + \frac{13}{5 (2)} \cdot 5$

Cancel the common factor.

$4 t^{2} = 11 t + \frac{13}{5 \cdot 2} \cdot 5$

Rewrite the expression.

$4 t^{2} = 11 t + \frac{13}{2}$

Step 3

Solve for $t$ .

Subtract $11 t$ from both sides of the equation.

$4 t^{2} - 11 t = \frac{13}{2}$

Subtract $\frac{13}{2}$ from both sides of the equation.

$4 t^{2} - 11 t - \frac{13}{2} = 0$

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

Apply the distributive property.

$2 (4 t^{2}) + 2 (- 11 t) + 2 (- \frac{13}{2}) = 0$

Simplify.

Multiply $4$ by $2$ .

$8 t^{2} + 2 (- 11 t) + 2 (- \frac{13}{2}) = 0$

Multiply $- 11$ by $2$ .

$8 t^{2} - 22 t + 2 (- \frac{13}{2}) = 0$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{13}{2}$ into the numerator.

$8 t^{2} - 22 t + 2 (\frac{- 13}{2}) = 0$

Cancel the common factor.

$8 t^{2} - 22 t + 2 (\frac{- 13}{2}) = 0$

Rewrite the expression.

$8 t^{2} - 22 t - 13 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 8$ , $b = - 22$ , and $c = - 13$ into the quadratic formula and solve for $t$ .

$\frac{22 \pm \sqrt{{(- 22)}^{2} - 4 \cdot (8 \cdot - 13)}}{2 \cdot 8}$

Simplify.

Simplify the numerator.

Raise $- 22$ to the power of $2$ .

$t = \frac{22 \pm \sqrt{484 - 4 \cdot 8 \cdot - 13}}{2 \cdot 8}$

Multiply $- 4 \cdot 8 \cdot - 13$ .

Multiply $- 4$ by $8$ .

$t = \frac{22 \pm \sqrt{484 - 32 \cdot - 13}}{2 \cdot 8}$

Multiply $- 32$ by $- 13$ .

$t = \frac{22 \pm \sqrt{484 + 416}}{2 \cdot 8}$

Add $484$ and $416$ .

$t = \frac{22 \pm \sqrt{900}}{2 \cdot 8}$

Rewrite $900$ as $30^{2}$ .

$t = \frac{22 \pm \sqrt{30^{2}}}{2 \cdot 8}$

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

$t = \frac{22 \pm 30}{2 \cdot 8}$

Multiply $2$ by $8$ .

$t = \frac{22 \pm 30}{16}$

Simplify $\frac{22 \pm 30}{16}$ .

$t = \frac{11 \pm 15}{8}$

The final answer is the combination of both solutions.

$t = \frac{13}{4}, - \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{13}{4}, - \frac{1}{2}$

Decimal Form:

$t = 3.25, - 0.5$

Mixed Number Form:

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

Solve for t (4t^2)/5=(11t)/5+13/10

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