Algebra Examples

$\sqrt{72 - p} = p$

Step 1

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

${\sqrt{72 - p}}^{2} = p^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{72 - p}$ as ${(72 - p)}^{\frac{1}{2}}$ .

${({(72 - p)}^{\frac{1}{2}})}^{2} = p^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(72 - p)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(72 - p)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(72 - p)}^{\frac{1}{2} \cdot 2} = p^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(72 - p)}^{\frac{1}{2} \cdot 2} = p^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(72 - p)}^{1} = p^{2}$

Step 2.2.1.2

Simplify.

$72 - p = p^{2}$

Step 3

Solve for $p$ .

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Step 3.1

Subtract $p^{2}$ from both sides of the equation.

$72 - p - p^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $72 - p - p^{2}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $72$ .

$- p - p^{2} + 72 = 0$

Step 3.2.1.1.2

Reorder $- p$ and $- p^{2}$ .

$- p^{2} - p + 72 = 0$

Step 3.2.1.2

Factor $- 1$ out of $- p^{2}$ .

$- (p^{2}) - p + 72 = 0$

Step 3.2.1.3

Factor $- 1$ out of $- p$ .

$- (p^{2}) - (p) + 72 = 0$

Step 3.2.1.4

Rewrite $72$ as $- 1 (- 72)$ .

$- (p^{2}) - (p) - 1 \cdot - 72 = 0$

Step 3.2.1.5

Factor $- 1$ out of $- (p^{2}) - (p)$ .

$- (p^{2} + p) - 1 \cdot - 72 = 0$

Step 3.2.1.6

Factor $- 1$ out of $- (p^{2} + p) - 1 (- 72)$ .

$- (p^{2} + p - 72) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $p^{2} + p - 72$ using the AC method.

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Step 3.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 72$ and whose sum is $1$ .

$- 8, 9$

Step 3.2.2.1.2

Write the factored form using these integers.

$- ((p - 8) (p + 9)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$- (p - 8) (p + 9) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$p - 8 = 0$

$p + 9 = 0$

Step 3.4

Set $p - 8$ equal to $0$ and solve for $p$ .

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Step 3.4.1

Set $p - 8$ equal to $0$ .

$p - 8 = 0$

Step 3.4.2

Add $8$ to both sides of the equation.

$p = 8$

Step 3.5

Set $p + 9$ equal to $0$ and solve for $p$ .

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Step 3.5.1

Set $p + 9$ equal to $0$ .

$p + 9 = 0$

Step 3.5.2

Subtract $9$ from both sides of the equation.

$p = - 9$

Step 3.6

The final solution is all the values that make $- (p - 8) (p + 9) = 0$ true.

$p = 8, - 9$

Step 4

Exclude the solutions that do not make $\sqrt{72 - p} = p$ true.

$p = 8$