Algebra Examples

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

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 15}$ as ${(2 x + 15)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $- {(2 x + 15)}^{\frac{1}{2}}$ .

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

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 3.2.1.3

Multiply ${({(2 x + 15)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

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

Step 3.2.1.4.2.2

Rewrite the expression.

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

Step 3.2.1.5

Simplify.

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

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

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $15$ .

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 4.2.2.1.1

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 $- 2$ .

$- 5, 3$

Step 4.2.2.1.2

Write the factored form using these integers.

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

Step 4.2.2.2

Remove unnecessary parentheses.

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

Step 4.3

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

$x - 5 = 0$

$x + 3 = 0$

Step 4.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.4.1

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

$x - 5 = 0$

Step 4.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.5

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

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Step 4.5.1

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

$x + 3 = 0$

Step 4.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.6

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

$x = 5, - 3$

Step 5

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

$x = - 3$