Algebra Examples

$5 = x + \sqrt{x + 1}$

Step 1

Rewrite the equation as $x + \sqrt{x + 1} = 5$ .

$x + \sqrt{x + 1} = 5$

Step 2

Subtract $x$ from both sides of the equation.

$\sqrt{x + 1} = 5 - x$

Step 3

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

${\sqrt{x + 1}}^{2} = {(5 - x)}^{2}$

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

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.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(5 - x)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(5 - x)}^{2}$ as $(5 - x) (5 - x)$ .

$x + 1 = (5 - x) (5 - x)$

Step 4.3.1.2

Expand $(5 - x) (5 - x)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$x + 1 = 5 (5 - x) - x (5 - x)$

Step 4.3.1.2.2

Apply the distributive property.

$x + 1 = 5 \cdot 5 + 5 (- x) - x (5 - x)$

Step 4.3.1.2.3

Apply the distributive property.

$x + 1 = 5 \cdot 5 + 5 (- x) - x \cdot 5 - x (- x)$

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

Multiply $5$ by $5$ .

$x + 1 = 25 + 5 (- x) - x \cdot 5 - x (- x)$

Step 4.3.1.3.1.2

Multiply $- 1$ by $5$ .

$x + 1 = 25 - 5 x - x \cdot 5 - x (- x)$

Step 4.3.1.3.1.3

Multiply $5$ by $- 1$ .

$x + 1 = 25 - 5 x - 5 x - x (- x)$

Step 4.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$x + 1 = 25 - 5 x - 5 x - 1 \cdot - 1 x \cdot x$

Step 4.3.1.3.1.5

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

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Step 4.3.1.3.1.5.1

Move $x$ .

$x + 1 = 25 - 5 x - 5 x - 1 \cdot - 1 (x \cdot x)$

Step 4.3.1.3.1.5.2

Multiply $x$ by $x$ .

$x + 1 = 25 - 5 x - 5 x - 1 \cdot - 1 x^{2}$

Step 4.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$x + 1 = 25 - 5 x - 5 x + 1 x^{2}$

Step 4.3.1.3.1.7

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

$x + 1 = 25 - 5 x - 5 x + x^{2}$

Step 4.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$x + 1 = 25 - 10 x + x^{2}$

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.

$25 - 10 x + x^{2} = x + 1$

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.

$25 - 10 x + x^{2} - x = 1$

Step 5.2.2

Subtract $x$ from $- 10 x$ .

$25 + x^{2} - 11 x = 1$

Step 5.3

Subtract $1$ from both sides of the equation.

$25 + x^{2} - 11 x - 1 = 0$

Step 5.4

Subtract $1$ from $25$ .

$x^{2} - 11 x + 24 = 0$

Step 5.5

Factor $x^{2} - 11 x + 24$ 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 $24$ and whose sum is $- 11$ .

$- 8, - 3$

Step 5.5.2

Write the factored form using these integers.

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

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 $0$ .

$x - 8 = 0$

$x - 3 = 0$

Step 5.7

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

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Step 5.7.1

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

$x - 8 = 0$

Step 5.7.2

Add $8$ to both sides of the equation.

$x = 8$

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 $(x - 8) (x - 3) = 0$ true.

$x = 8, 3$

Step 6

Exclude the solutions that do not make $5 = x + \sqrt{x + 1}$ true.

$x = 3$

Step 7