Solve by Factoring square root of x+10-1=x

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

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{x + 10} - 1 - x = 0$

Step 2

Move all terms not containing $\sqrt{x + 10}$ to the right side of the equation.

Add $1$ to both sides of the equation.

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

Add $x$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 10}$ as ${(x + 10)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $x$ .

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

Add $x$ and $x$ .

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

Step 5

Solve for $x$ .

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

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $2 x$ .

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

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

Subtract $10$ from both sides of the equation.

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

Subtract $10$ from $1$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 9$ .

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

Add $1$ and $36$ .

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

Multiply $2$ by $1$ .

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

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 9$ .

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

Add $1$ and $36$ .

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

Multiply $2$ by $1$ .

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

Change the $\pm$ to $+$ .

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

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

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

Factor $- 1$ out of $\sqrt{37}$ .

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

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{37})$ .

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

Move the negative in front of the fraction.

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

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 9$ .

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

Add $1$ and $36$ .

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

Multiply $2$ by $1$ .

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

Change the $\pm$ to $-$ .

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

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

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

Factor $- 1$ out of $- \sqrt{37}$ .

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

Factor $- 1$ out of $- 1 (1) - (\sqrt{37})$ .

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

Move the negative in front of the fraction.

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

The final answer is the combination of both solutions.

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

Step 6

Exclude the solutions that do not make $\sqrt{x + 10} - 1 - x = 0$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.54138126 \dots$

Solve by Factoring square root of x+10-1=x

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