Algebra Examples

$\sqrt{x + 10} - 1 = x$

Step 1

Add $1$ to both sides of the equation.

$\sqrt{x + 10} = x + 1$

Step 2

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

${\sqrt{x + 10}}^{2} = {(x + 1)}^{2}$

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 $\sqrt{x + 10}$ as ${(x + 10)}^{\frac{1}{2}}$ .

${({(x + 10)}^{\frac{1}{2}})}^{2} = {(x + 1)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x + 10)}^{\frac{1}{2} \cdot 2} = {(x + 1)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x + 10)}^{\frac{1}{2} \cdot 2} = {(x + 1)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 10)}^{1} = {(x + 1)}^{2}$

Step 3.2.1.2

Simplify.

$x + 10 = {(x + 1)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$x + 10 = (x + 1) (x + 1)$

Step 3.3.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x + 10 = x (x + 1) + 1 (x + 1)$

Step 3.3.1.2.2

Apply the distributive property.

$x + 10 = x \cdot x + x \cdot 1 + 1 (x + 1)$

Step 3.3.1.2.3

Apply the distributive property.

$x + 10 = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $x$ by $x$ .

$x + 10 = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Step 3.3.1.3.1.2

Multiply $x$ by $1$ .

$x + 10 = x^{2} + x + 1 x + 1 \cdot 1$

Step 3.3.1.3.1.3

Multiply $x$ by $1$ .

$x + 10 = x^{2} + x + x + 1 \cdot 1$

Step 3.3.1.3.1.4

Multiply $1$ by $1$ .

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

Step 3.3.1.3.2

Add $x$ and $x$ .

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Subtract $x$ from both sides of the equation.

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

Step 4.2.2

Subtract $x$ from $2 x$ .

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

Step 4.3

Move all terms to the left side of the equation and simplify.

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Step 4.3.1

Subtract $10$ from both sides of the equation.

$x^{2} + x + 1 - 10 = 0$

Step 4.3.2

Subtract $10$ from $1$ .

$x^{2} + x - 9 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5

Substitute the values $a = 1$ , $b = 1$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 9}}{2 \cdot 1}$

Step 4.6.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 1 \pm \sqrt{1 + 36}}{2 \cdot 1}$

Step 4.6.1.3

Add $1$ and $36$ .

$x = \frac{- 1 \pm \sqrt{37}}{2 \cdot 1}$

Step 4.6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{37}}{2}$

Step 4.7

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{37}}{2}, - \frac{1 + \sqrt{37}}{2}$

Step 5

Exclude the solutions that do not make $\sqrt{x + 10} - 1 = x$ true.

$x = - \frac{1 - \sqrt{37}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{37}}{2}$

Decimal Form:

$x = 2.54138126 \dots$