Pre-Algebra Examples

$\sqrt{24 - 5 x} = x$

Step 1

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

${\sqrt{24 - 5 x}}^{2} = x^{2}$

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 $\sqrt{24 - 5 x}$ as ${(24 - 5 x)}^{\frac{1}{2}}$ .

${({(24 - 5 x)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(24 - 5 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(24 - 5 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(24 - 5 x)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(24 - 5 x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(24 - 5 x)}^{1} = x^{2}$

Step 2.2.1.2

Simplify.

$24 - 5 x = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

$24 - 5 x - x^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $24 - 5 x - x^{2}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $24$ .

$- 5 x - x^{2} + 24 = 0$

Step 3.2.1.1.2

Reorder $- 5 x$ and $- x^{2}$ .

$- x^{2} - 5 x + 24 = 0$

Step 3.2.1.2

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

$- (x^{2}) - 5 x + 24 = 0$

Step 3.2.1.3

Factor $- 1$ out of $- 5 x$ .

$- (x^{2}) - (5 x) + 24 = 0$

Step 3.2.1.4

Rewrite $24$ as $- 1 (- 24)$ .

$- (x^{2}) - (5 x) - 1 \cdot - 24 = 0$

Step 3.2.1.5

Factor $- 1$ out of $- (x^{2}) - (5 x)$ .

$- (x^{2} + 5 x) - 1 \cdot - 24 = 0$

Step 3.2.1.6

Factor $- 1$ out of $- (x^{2} + 5 x) - 1 (- 24)$ .

$- (x^{2} + 5 x - 24) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 3.2.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 8)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 8) = 0$

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 $0$ .

$x - 3 = 0$

$x + 8 = 0$

Step 3.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.5

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

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Step 3.5.1

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

$x + 8 = 0$

Step 3.5.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3.6

The final solution is all the values that make $- (x - 3) (x + 8) = 0$ true.

$x = 3, - 8$

Step 4

Exclude the solutions that do not make $\sqrt{24 - 5 x} = x$ true.

$x = 3$