Algebra Examples

$\sqrt{x + 72} = x$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{x + 72} - x = 0$

Step 2

Add $x$ to both sides of the equation.

$\sqrt{x + 72} = x$

Step 3

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

${\sqrt{x + 72}}^{2} = x^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${({(x + 72)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(x + 72)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

${(x + 72)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(x + 72)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(x + 72)}^{1} = x^{2}$

Step 4.2.1.2

Simplify.

$x + 72 = x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

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

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

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

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

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Step 5.2.1.1

Reorder the expression.

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Step 5.2.1.1.1

Move $72$ .

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

Step 5.2.1.1.2

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

Factor $- 1$ out of $x$ .

$- (x^{2}) - 1 (- x) + 72 = 0$

Step 5.2.1.4

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

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

Step 5.2.1.5

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

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

Step 5.2.1.6

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

$- (x^{2} - x - 72) = 0$

Step 5.2.2

Factor.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

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 $- 1$ .

$- 9, 8$

Step 5.2.2.1.2

Write the factored form using these integers.

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

Step 5.2.2.2

Remove unnecessary parentheses.

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

Step 5.3

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

$x - 9 = 0$

$x + 8 = 0$

Step 5.4

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 5.4.2

Add $9$ to both sides of the equation.

$x = 9$

Step 5.5

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 5.5.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 5.6

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

$x = 9, - 8$

Step 6

Exclude the solutions that do not make $\sqrt{x + 72} - x = 0$ true.

$x = 9$