Write as a Function of d d=144-16t^2

$d = 144 - 16 t^{2}$

Step 1

Rewrite the equation as $144 - 16 t^{2} = d$ .

$144 - 16 t^{2} = d$

Step 2

Subtract $144$ from both sides of the equation.

$- 16 t^{2} = d - 144$

Step 3

Divide each term in $- 16 t^{2} = d - 144$ by $- 16$ and simplify.

Divide each term in $- 16 t^{2} = d - 144$ by $- 16$ .

$\frac{- 16 t^{2}}{- 16} = \frac{d}{- 16} + \frac{- 144}{- 16}$

Simplify the left side.

Cancel the common factor of $- 16$ .

Cancel the common factor.

$\frac{- 16 t^{2}}{- 16} = \frac{d}{- 16} + \frac{- 144}{- 16}$

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

$t^{2} = \frac{d}{- 16} + \frac{- 144}{- 16}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$t^{2} = - \frac{d}{16} + \frac{- 144}{- 16}$

Divide $- 144$ by $- 16$ .

$t^{2} = - \frac{d}{16} + 9$

Step 4

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

$t = \pm \sqrt{- \frac{d}{16} + 9}$

Step 5

Simplify $\pm \sqrt{- \frac{d}{16} + 9}$ .

To write $9$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$t = \pm \sqrt{- \frac{d}{16} + 9 \cdot \frac{16}{16}}$

Combine $9$ and $\frac{16}{16}$ .

$t = \pm \sqrt{- \frac{d}{16} + \frac{9 \cdot 16}{16}}$

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{- d + 9 \cdot 16}{16}}$

Multiply $9$ by $16$ .

$t = \pm \sqrt{\frac{- d + 144}{16}}$

Rewrite $\frac{- d + 144}{16}$ as ${(\frac{1}{4})}^{2} (- d + 144)$ .

Factor the perfect power $1^{2}$ out of $- d + 144$ .

$t = \pm \sqrt{\frac{1^{2} (- d + 144)}{16}}$

Factor the perfect power $4^{2}$ out of $16$ .

$t = \pm \sqrt{\frac{1^{2} (- d + 144)}{4^{2} \cdot 1}}$

Rearrange the fraction $\frac{1^{2} (- d + 144)}{4^{2} \cdot 1}$ .

$t = \pm \sqrt{{(\frac{1}{4})}^{2} (- d + 144)}$

Pull terms out from under the radical.

$t = \pm \frac{1}{4} \sqrt{- d + 144}$

Combine $\frac{1}{4}$ and $\sqrt{- d + 144}$ .

$t = \pm \frac{\sqrt{- d + 144}}{4}$

Step 6

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

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

$t = \frac{\sqrt{- d + 144}}{4}$

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

$t = - \frac{\sqrt{- d + 144}}{4}$

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

$t = \frac{\sqrt{- d + 144}}{4}$

$t = - \frac{\sqrt{- d + 144}}{4}$

$t = \frac{\sqrt{- d + 144}}{4}$

$t = - \frac{\sqrt{- d + 144}}{4}$

Step 7

To rewrite as a function of $d$ , write the equation so that $t$ is by itself on one side of the equal sign and an expression involving only $d$ is on the other side.

$f (d) = \frac{\sqrt{- d + 144}}{4}, - \frac{\sqrt{- d + 144}}{4}$

Write as a Function of d d=144-16t^2

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