Algebra Examples

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

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as ${(- x - 1)}^{2} + 3 = 0$ .

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

Step 1.2.2

Subtract $3$ from both sides of the equation.

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

Step 1.2.3

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

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

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

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

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

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

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

Step 1.2.5.2

Add $1$ to both sides of the equation.

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

Step 1.2.5.3

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

Tap for more steps...

Step 1.2.5.3.1

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

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

Step 1.2.5.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.5.3.2.2

Divide $x$ by $1$ .

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

Step 1.2.5.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.3.3.1

Simplify each term.

Tap for more steps...

Step 1.2.5.3.3.1.1

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

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

Step 1.2.5.3.3.1.2

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

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

Step 1.2.5.3.3.1.3

Divide $1$ by $- 1$ .

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

Step 1.2.5.4

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

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

Step 1.2.5.5

Add $1$ to both sides of the equation.

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

Step 1.2.5.6

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

Tap for more steps...

Step 1.2.5.6.1

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

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

Step 1.2.5.6.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.6.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.5.6.2.2

Divide $x$ by $1$ .

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

Step 1.2.5.6.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.6.3.1

Simplify each term.

Tap for more steps...

Step 1.2.5.6.3.1.1

Dividing two negative values results in a positive value.

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

Step 1.2.5.6.3.1.2

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

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

Step 1.2.5.6.3.1.3

Divide $1$ by $- 1$ .

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

Step 1.2.5.7

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 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Simplify ${(- (0) - 1)}^{2} + 3$ .

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Multiply $- 1$ by $0$ .

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

Step 2.2.2.1.2

Subtract $1$ from $0$ .

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

Step 2.2.2.1.3

Raise $- 1$ to the power of $2$ .

$y = 1 + 3$

Step 2.2.2.2

Add $1$ and $3$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 4)$

Step 4