Solve for r r = square root of -1-2r

$r = \sqrt{- 1 - 2 r}$

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.

$\sqrt{- 1 - 2 r} = r$

Step 2

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

${\sqrt{- 1 - 2 r}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

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

${({(- 1 - 2 r)}^{\frac{1}{2}})}^{2} = r^{2}$

Simplify the left side.

Simplify ${({(- 1 - 2 r)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- 1 - 2 r)}^{\frac{1}{2}})}^{2}$ .

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

${(- 1 - 2 r)}^{\frac{1}{2} \cdot 2} = r^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 1 - 2 r)}^{\frac{1}{2} \cdot 2} = r^{2}$

Rewrite the expression.

${(- 1 - 2 r)}^{1} = r^{2}$

Simplify.

$- 1 - 2 r = r^{2}$

Step 4

Solve for $r$ .

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

$- 1 - 2 r - r^{2} = 0$

Factor the left side of the equation.

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

Reorder the expression.

Move $- 1$ .

$- 2 r - r^{2} - 1 = 0$

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

$- r^{2} - 2 r - 1 = 0$

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

$- (r^{2}) - 2 r - 1 = 0$

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

$- (r^{2}) - (2 r) - 1 = 0$

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

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

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

$- (r^{2} + 2 r) - 1 \cdot 1 = 0$

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

$- (r^{2} + 2 r + 1) = 0$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$- (r^{2} + 2 r + 1^{2}) = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 r = 2 \cdot r \cdot 1$

Rewrite the polynomial.

$- (r^{2} + 2 \cdot r \cdot 1 + 1^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = r$ and $b = 1$ .

$- {(r + 1)}^{2} = 0$

Divide each term in $- {(r + 1)}^{2} = 0$ by $- 1$ and simplify.

Divide each term in $- {(r + 1)}^{2} = 0$ by $- 1$ .

$\frac{- {(r + 1)}^{2}}{- 1} = \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{{(r + 1)}^{2}}{1} = \frac{0}{- 1}$

Divide ${(r + 1)}^{2}$ by $1$ .

${(r + 1)}^{2} = \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

${(r + 1)}^{2} = 0$

Set the $r + 1$ equal to $0$ .

$r + 1 = 0$

Subtract $1$ from both sides of the equation.

$r = - 1$

Step 5

Exclude the solutions that do not make $r = \sqrt{- 1 - 2 r}$ true.

$No solution$

Solve for r r = square root of -1-2r

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