Algebra Examples

$y = {(- x - 1)}^{2} + 3$

Step 1

Set ${(- x - 1)}^{2} + 3$ equal to $0$ .

${(- x - 1)}^{2} + 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $3$ from both sides of the equation.

${(- x - 1)}^{2} = - 3$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$- x - 1 = \pm \sqrt{- 3}$

Step 2.3

Simplify $\pm \sqrt{- 3}$ .

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Step 2.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$- x - 1 = \pm \sqrt{- 1 (3)}$

Step 2.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$- x - 1 = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$- x - 1 = \pm i \sqrt{3}$

Step 2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$- x - 1 = i \sqrt{3}$

Step 2.4.2

Add $1$ to both sides of the equation.

$- x = i \sqrt{3} + 1$

Step 2.4.3

Divide each term in $- x = i \sqrt{3} + 1$ by $- 1$ and simplify.

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Step 2.4.3.1

Divide each term in $- x = i \sqrt{3} + 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Step 2.4.3.2.2

Divide $x$ by $1$ .

$x = \frac{i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Simplify each term.

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Step 2.4.3.3.1.1

Move the negative one from the denominator of $\frac{i \sqrt{3}}{- 1}$ .

$x = - 1 \cdot (i \sqrt{3}) + \frac{1}{- 1}$

Step 2.4.3.3.1.2

Rewrite $- 1 \cdot (i \sqrt{3})$ as $- (i \sqrt{3})$ .

$x = - (i \sqrt{3}) + \frac{1}{- 1}$

Step 2.4.3.3.1.3

Divide $1$ by $- 1$ .

$x = - i \sqrt{3} - 1$

Step 2.4.4

Next, use the negative value of the $\pm$ to find the second solution.

$- x - 1 = - i \sqrt{3}$

Step 2.4.5

Add $1$ to both sides of the equation.

$- x = - i \sqrt{3} + 1$

Step 2.4.6

Divide each term in $- x = - i \sqrt{3} + 1$ by $- 1$ and simplify.

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Step 2.4.6.1

Divide each term in $- x = - i \sqrt{3} + 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Step 2.4.6.2

Simplify the left side.

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Step 2.4.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Step 2.4.6.2.2

Divide $x$ by $1$ .

$x = \frac{- i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Step 2.4.6.3

Simplify the right side.

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Step 2.4.6.3.1

Simplify each term.

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Step 2.4.6.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{i \sqrt{3}}{1} + \frac{1}{- 1}$

Step 2.4.6.3.1.2

Divide $i \sqrt{3}$ by $1$ .

$x = i \sqrt{3} + \frac{1}{- 1}$

Step 2.4.6.3.1.3

Divide $1$ by $- 1$ .

$x = i \sqrt{3} - 1$

Step 2.4.7

The complete solution is the result of both the positive and negative portions of the solution.

$x = - i \sqrt{3} - 1, i \sqrt{3} - 1$

Step 3