Algebra Examples

$\sqrt{15 - 2 x} = x$

Step 1

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

${\sqrt{15 - 2 x}}^{2} = x^{2}$

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 $\sqrt{15 - 2 x}$ as ${(15 - 2 x)}^{\frac{1}{2}}$ .

${({(15 - 2 x)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(15 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(15 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(15 - 2 x)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(15 - 2 x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(15 - 2 x)}^{1} = x^{2}$

Step 2.2.1.2

Simplify.

$15 - 2 x = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

$15 - 2 x - x^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $15 - 2 x - x^{2}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $15$ .

$- 2 x - x^{2} + 15 = 0$

Step 3.2.1.1.2

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

$- x^{2} - 2 x + 15 = 0$

Step 3.2.1.2

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

$- (x^{2}) - 2 x + 15 = 0$

Step 3.2.1.3

Factor $- 1$ out of $- 2 x$ .

$- (x^{2}) - (2 x) + 15 = 0$

Step 3.2.1.4

Rewrite $15$ as $- 1 (- 15)$ .

$- (x^{2}) - (2 x) - 1 \cdot - 15 = 0$

Step 3.2.1.5

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

$- (x^{2} + 2 x) - 1 \cdot - 15 = 0$

Step 3.2.1.6

Factor $- 1$ out of $- (x^{2} + 2 x) - 1 (- 15)$ .

$- (x^{2} + 2 x - 15) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} + 2 x - 15$ 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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 3.2.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 5)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 5) = 0$

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 $0$ .

$x - 3 = 0$

$x + 5 = 0$

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 $x + 5$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 3.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3.6

The final solution is all the values that make $- (x - 3) (x + 5) = 0$ true.

$x = 3, - 5$

Step 4

Exclude the solutions that do not make $\sqrt{15 - 2 x} = x$ true.

$x = 3$