Precalculus Examples

$h (t) = \sqrt{9 - t^{2}}$

Step 1

Set $\sqrt{9 - t^{2}}$ equal to $0$ .

$\sqrt{9 - t^{2}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $\sqrt{9 - t^{2}}$ .

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Step 2.1.1

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

$\sqrt{3^{2} - t^{2}} = 0$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = t$ .

$\sqrt{(3 + t) (3 - t)} = 0$

Step 2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{(3 + t) (3 - t)}}^{2} = 0^{2}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(3 + t) (3 - t)}$ as ${((3 + t) (3 - t))}^{\frac{1}{2}}$ .

${({((3 + t) (3 - t))}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({((3 + t) (3 - t))}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({((3 + t) (3 - t))}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((3 + t) (3 - t))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${((3 + t) (3 - t))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${((3 + t) (3 - t))}^{1} = 0^{2}$

Step 2.3.2.1.2

Expand $(3 + t) (3 - t)$ using the FOIL Method.

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Step 2.3.2.1.2.1

Apply the distributive property.

${(3 (3 - t) + t (3 - t))}^{1} = 0^{2}$

Step 2.3.2.1.2.2

Apply the distributive property.

${(3 \cdot 3 + 3 (- t) + t (3 - t))}^{1} = 0^{2}$

Step 2.3.2.1.2.3

Apply the distributive property.

${(3 \cdot 3 + 3 (- t) + t \cdot 3 + t (- t))}^{1} = 0^{2}$

Step 2.3.2.1.3

Simplify and combine like terms.

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Step 2.3.2.1.3.1

Simplify each term.

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Step 2.3.2.1.3.1.1

Multiply $3$ by $3$ .

${(9 + 3 (- t) + t \cdot 3 + t (- t))}^{1} = 0^{2}$

Step 2.3.2.1.3.1.2

Multiply $- 1$ by $3$ .

${(9 - 3 t + t \cdot 3 + t (- t))}^{1} = 0^{2}$

Step 2.3.2.1.3.1.3

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

${(9 - 3 t + 3 \cdot t + t (- t))}^{1} = 0^{2}$

Step 2.3.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(9 - 3 t + 3 t - t \cdot t)}^{1} = 0^{2}$

Step 2.3.2.1.3.1.5

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

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Step 2.3.2.1.3.1.5.1

Move $t$ .

${(9 - 3 t + 3 t - (t \cdot t))}^{1} = 0^{2}$

Step 2.3.2.1.3.1.5.2

Multiply $t$ by $t$ .

${(9 - 3 t + 3 t - t^{2})}^{1} = 0^{2}$

Step 2.3.2.1.3.2

Add $- 3 t$ and $3 t$ .

${(9 + 0 - t^{2})}^{1} = 0^{2}$

Step 2.3.2.1.3.3

Add $9$ and $0$ .

${(9 - t^{2})}^{1} = 0^{2}$

Step 2.3.2.1.4

Simplify.

$9 - t^{2} = 0^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Raising $0$ to any positive power yields $0$ .

$9 - t^{2} = 0$

Step 2.4

Solve for $t$ .

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Step 2.4.1

Subtract $9$ from both sides of the equation.

$- t^{2} = - 9$

Step 2.4.2

Divide each term in $- t^{2} = - 9$ by $- 1$ and simplify.

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Step 2.4.2.1

Divide each term in $- t^{2} = - 9$ by $- 1$ .

$\frac{- t^{2}}{- 1} = \frac{- 9}{- 1}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{t^{2}}{1} = \frac{- 9}{- 1}$

Step 2.4.2.2.2

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

$t^{2} = \frac{- 9}{- 1}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 9$ by $- 1$ .

$t^{2} = 9$

Step 2.4.3

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

$t = \pm \sqrt{9}$

Step 2.4.4

Simplify $\pm \sqrt{9}$ .

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Step 2.4.4.1

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

$t = \pm \sqrt{3^{2}}$

Step 2.4.4.2

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

$t = \pm 3$

Step 2.4.5

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

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Step 2.4.5.1

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

$t = 3$

Step 2.4.5.2

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

$t = - 3$

Step 2.4.5.3

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

$t = 3, - 3$

Step 3