Finite Math Examples

$0 = - 4.9 t^{2} + 80 t$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 4.9 t^{2} + 80 t = 0$

Step 2

Divide each term in $- 4.9 t^{2} + 80 t = 0$ by $- 4.9$ and simplify.

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Step 2.1

Divide each term in $- 4.9 t^{2} + 80 t = 0$ by $- 4.9$ .

$\frac{- 4.9 t^{2}}{- 4.9} + \frac{80 t}{- 4.9} = \frac{0}{- 4.9}$

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

Cancel the common factor of $- 4.9$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{- 4.9 t^{2}}{- 4.9} + \frac{80 t}{- 4.9} = \frac{0}{- 4.9}$

Step 2.2.1.1.2

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

$t^{2} + \frac{80 t}{- 4.9} = \frac{0}{- 4.9}$

Step 2.2.1.2

Move the negative in front of the fraction.

$t^{2} - \frac{80 t}{4.9} = \frac{0}{- 4.9}$

Step 2.2.1.3

Factor $80$ out of $80 t$ .

$t^{2} - \frac{80 (t)}{4.9} = \frac{0}{- 4.9}$

Step 2.2.1.4

Factor $4.9$ out of $4.9$ .

$t^{2} - \frac{80 (t)}{4.9 (1)} = \frac{0}{- 4.9}$

Step 2.2.1.5

Separate fractions.

$t^{2} - (\frac{80}{4.9} \cdot \frac{t}{1}) = \frac{0}{- 4.9}$

Step 2.2.1.6

Divide $80$ by $4.9$ .

$t^{2} - (16.32653061 \frac{t}{1}) = \frac{0}{- 4.9}$

Step 2.2.1.7

Divide $t$ by $1$ .

$t^{2} - (16.32653061 t) = \frac{0}{- 4.9}$

Step 2.2.1.8

Multiply $16.32653061$ by $- 1$ .

$t^{2} - 16.32653061 t = \frac{0}{- 4.9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 4.9$ .

$t^{2} - 16.32653061 t = 0$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{400}{49})}^{2}$

Step 4

Add the term to each side of the equation.

$t^{2} - 16.32653061 t + {(- \frac{400}{49})}^{2} = 0 + {(- \frac{400}{49})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

Apply the product rule to $- \frac{400}{49}$ .

$t^{2} - 16.32653061 t + {(- 1)}^{2} {(\frac{400}{49})}^{2} = 0 + {(- \frac{400}{49})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{400}{49}$ .

$t^{2} - 16.32653061 t + {(- 1)}^{2} \frac{400^{2}}{49^{2}} = 0 + {(- \frac{400}{49})}^{2}$

Step 5.1.1.2

Raise $- 1$ to the power of $2$ .

$t^{2} - 16.32653061 t + 1 \frac{400^{2}}{49^{2}} = 0 + {(- \frac{400}{49})}^{2}$

Step 5.1.1.3

Multiply $\frac{400^{2}}{49^{2}}$ by $1$ .

$t^{2} - 16.32653061 t + \frac{400^{2}}{49^{2}} = 0 + {(- \frac{400}{49})}^{2}$

Step 5.1.1.4

Raise $400$ to the power of $2$ .

$t^{2} - 16.32653061 t + \frac{160000}{49^{2}} = 0 + {(- \frac{400}{49})}^{2}$

Step 5.1.1.5

Raise $49$ to the power of $2$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + {(- \frac{400}{49})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $0 + {(- \frac{400}{49})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{400}{49}$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + {(- 1)}^{2} {(\frac{400}{49})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{400}{49}$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + {(- 1)}^{2} \frac{400^{2}}{49^{2}}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + 1 \frac{400^{2}}{49^{2}}$

Step 5.2.1.1.3

Multiply $\frac{400^{2}}{49^{2}}$ by $1$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + \frac{400^{2}}{49^{2}}$

Step 5.2.1.1.4

Raise $400$ to the power of $2$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + \frac{160000}{49^{2}}$

Step 5.2.1.1.5

Raise $49$ to the power of $2$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = 0 + \frac{160000}{2401}$

Step 5.2.1.2

Add $0$ and $\frac{160000}{2401}$ .

$t^{2} - 16.32653061 t + \frac{160000}{2401} = \frac{160000}{2401}$

Step 6

Factor the perfect trinomial square into ${(t - \frac{400}{49})}^{2}$ .

${(t - \frac{400}{49})}^{2} = \frac{160000}{2401}$

Step 7

Solve the equation for $t$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t - \frac{400}{49} = \pm \sqrt{\frac{160000}{2401}}$

Step 7.2

Simplify $\pm \sqrt{\frac{160000}{2401}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{160000}{2401}}$ as $\frac{\sqrt{160000}}{\sqrt{2401}}$ .

$t - \frac{400}{49} = \pm \frac{\sqrt{160000}}{\sqrt{2401}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $160000$ as $400^{2}$ .

$t - \frac{400}{49} = \pm \frac{\sqrt{400^{2}}}{\sqrt{2401}}$

Step 7.2.2.2

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

$t - \frac{400}{49} = \pm \frac{400}{\sqrt{2401}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $2401$ as $49^{2}$ .

$t - \frac{400}{49} = \pm \frac{400}{\sqrt{49^{2}}}$

Step 7.2.3.2

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

$t - \frac{400}{49} = \pm \frac{400}{49}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$t - \frac{400}{49} = \frac{400}{49}$

Step 7.3.2

Move all terms not containing $t$ to the right side of the equation.

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Step 7.3.2.1

Add $\frac{400}{49}$ to both sides of the equation.

$t = \frac{400}{49} + \frac{400}{49}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$t = \frac{400 + 400}{49}$

Step 7.3.2.3

Add $400$ and $400$ .

$t = \frac{800}{49}$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$t - \frac{400}{49} = - \frac{400}{49}$

Step 7.3.4

Move all terms not containing $t$ to the right side of the equation.

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Step 7.3.4.1

Add $\frac{400}{49}$ to both sides of the equation.

$t = - \frac{400}{49} + \frac{400}{49}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$t = \frac{- 400 + 400}{49}$

Step 7.3.4.3

Add $- 400$ and $400$ .

$t = \frac{0}{49}$

Step 7.3.4.4

Divide $0$ by $49$ .

$t = 0$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{800}{49}, 0$