Algebra Examples

$f (x) = - (x^{2} + 1) (x - 4)$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = - (x^{2} + 1) (x - 4)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- (x^{2} + 1) (x - 4) = 0$ .

$- (x^{2} + 1) (x - 4) = 0$

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} + 1 = 0$

$x - 4 = 0$

Step 1.2.3

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 1.2.3.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.3.2.2

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

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

Step 1.2.3.2.3

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

$x = \pm i$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

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

$x = i$

Step 1.2.3.2.4.2

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

$x = - i$

Step 1.2.3.2.4.3

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

$x = i, - i$

Step 1.2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 1.2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.5

The final solution is all the values that make $- (x^{2} + 1) (x - 4) = 0$ true.

$x = i, - i, 4$

Step 1.3

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = - ({(0)}^{2} + 1) ((0) - 4)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - (0^{2} + 1) ((0) - 4)$

Step 2.2.2

Remove parentheses.

$y = - (0^{2} + 1) (0 - 4)$

Step 2.2.3

Remove parentheses.

$y = - ({(0)}^{2} + 1) ((0) - 4)$

Step 2.2.4

Simplify $- ({(0)}^{2} + 1) ((0) - 4)$ .

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Step 2.2.4.1

Raising $0$ to any positive power yields $0$ .

$y = - (0 + 1) ((0) - 4)$

Step 2.2.4.2

Add $0$ and $1$ .

$y = - 1 \cdot 1 ((0) - 4)$

Step 2.2.4.3

Multiply $- 1$ by $1$ .

$y = - 1 ((0) - 4)$

Step 2.2.4.4

Subtract $4$ from $0$ .

$y = - 1 \cdot - 4$

Step 2.2.4.5

Multiply $- 1$ by $- 4$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4