Algebra Examples

$x = \sqrt{x + 13} + 7$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\sqrt{x + 13}$ from both sides of the equation.

$x - \sqrt{x + 13} = 7$

Step 1.2

Subtract $7$ from both sides of the equation.

$x - \sqrt{x + 13} - 7 = 0$

Step 2

Move all terms not containing $- \sqrt{x + 13}$ to the right side of the equation.

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Step 2.1

Subtract $x$ from both sides of the equation.

$- \sqrt{x + 13} - 7 = - x$

Step 2.2

Add $7$ to both sides of the equation.

$- \sqrt{x + 13} = - x + 7$

Step 3

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

${(- \sqrt{x + 13})}^{2} = {(- x + 7)}^{2}$

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 $\sqrt{x + 13}$ as ${(x + 13)}^{\frac{1}{2}}$ .

${(- {(x + 13)}^{\frac{1}{2}})}^{2} = {(- x + 7)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(- {(x + 13)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $- {(x + 13)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x + 13)}^{\frac{1}{2}})}^{2} = {(- x + 7)}^{2}$

Step 4.2.1.2

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

$1 {({(x + 13)}^{\frac{1}{2}})}^{2} = {(- x + 7)}^{2}$

Step 4.2.1.3

Multiply ${({(x + 13)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x + 13)}^{\frac{1}{2}})}^{2} = {(- x + 7)}^{2}$

Step 4.2.1.4

Multiply the exponents in ${({(x + 13)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

${(x + 13)}^{\frac{1}{2} \cdot 2} = {(- x + 7)}^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

${(x + 13)}^{\frac{1}{2} \cdot 2} = {(- x + 7)}^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

${(x + 13)}^{1} = {(- x + 7)}^{2}$

Step 4.2.1.5

Simplify.

$x + 13 = {(- x + 7)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(- x + 7)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(- x + 7)}^{2}$ as $(- x + 7) (- x + 7)$ .

$x + 13 = (- x + 7) (- x + 7)$

Step 4.3.1.2

Expand $(- x + 7) (- x + 7)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$x + 13 = - x (- x + 7) + 7 (- x + 7)$

Step 4.3.1.2.2

Apply the distributive property.

$x + 13 = - x (- x) - x \cdot 7 + 7 (- x + 7)$

Step 4.3.1.2.3

Apply the distributive property.

$x + 13 = - x (- x) - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$x + 13 = - 1 \cdot - 1 x \cdot x - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.3.1.2.1

Move $x$ .

$x + 13 = - 1 \cdot - 1 (x \cdot x) - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3.1.2.2

Multiply $x$ by $x$ .

$x + 13 = - 1 \cdot - 1 x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$x + 13 = 1 x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$x + 13 = x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3.1.5

Multiply $7$ by $- 1$ .

$x + 13 = x^{2} - 7 x + 7 (- x) + 7 \cdot 7$

Step 4.3.1.3.1.6

Multiply $- 1$ by $7$ .

$x + 13 = x^{2} - 7 x - 7 x + 7 \cdot 7$

Step 4.3.1.3.1.7

Multiply $7$ by $7$ .

$x + 13 = x^{2} - 7 x - 7 x + 49$

Step 4.3.1.3.2

Subtract $7 x$ from $- 7 x$ .

$x + 13 = x^{2} - 14 x + 49$

Step 5

Solve for $x$ .

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Step 5.1

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

$x^{2} - 14 x + 49 = x + 13$

Step 5.2

Move all terms containing $x$ to the left side of the equation.

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Step 5.2.1

Subtract $x$ from both sides of the equation.

$x^{2} - 14 x + 49 - x = 13$

Step 5.2.2

Subtract $x$ from $- 14 x$ .

$x^{2} - 15 x + 49 = 13$

Step 5.3

Subtract $13$ from both sides of the equation.

$x^{2} - 15 x + 49 - 13 = 0$

Step 5.4

Subtract $13$ from $49$ .

$x^{2} - 15 x + 36 = 0$

Step 5.5

Factor $x^{2} - 15 x + 36$ using the AC method.

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Step 5.5.1

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 $36$ and whose sum is $- 15$ .

$- 12, - 3$

Step 5.5.2

Write the factored form using these integers.

$(x - 12) (x - 3) = 0$

Step 5.6

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

$x - 12 = 0$

$x - 3 = 0$

Step 5.7

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 5.7.2

Add $12$ to both sides of the equation.

$x = 12$

Step 5.8

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

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Step 5.8.1

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

$x - 3 = 0$

Step 5.8.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5.9

The final solution is all the values that make $(x - 12) (x - 3) = 0$ true.

$x = 12, 3$

Step 6

Exclude the solutions that do not make $x - \sqrt{x + 13} - 7 = 0$ true.

$x = 12$