Find the x and y Intercepts f(x)=-2(x^2+3)(2x-4)(3x+1)^2

$f (x) = - 2 (x^{2} + 3) (2 x - 4) {(3 x + 1)}^{2}$

Step 1

Find the x-intercepts.

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

$0 = - 2 (x^{2} + 3) (2 x - 4) {(3 x + 1)}^{2}$

Solve the equation.

Rewrite the equation as $- 2 (x^{2} + 3) (2 x - 4) {(3 x + 1)}^{2} = 0$ .

$- 2 (x^{2} + 3) (2 x - 4) {(3 x + 1)}^{2} = 0$

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

$x^{2} + 3 = 0$

$2 x - 4 = 0$

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

Set $x^{2} + 3$ equal to $0$ and solve for $x$ .

Set $x^{2} + 3$ equal to $0$ .

$x^{2} + 3 = 0$

Solve $x^{2} + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

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

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

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

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

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

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

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

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

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

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

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

$x = i \sqrt{3}$

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

$x = - i \sqrt{3}$

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

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

Set $2 x - 4$ equal to $0$ and solve for $x$ .

Set $2 x - 4$ equal to $0$ .

$2 x - 4 = 0$

Solve $2 x - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

$2 x = 4$

Divide each term in $2 x = 4$ by $2$ and simplify.

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x = 2$

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

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

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

Solve ${(3 x + 1)}^{2} = 0$ for $x$ .

Set the $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

The final solution is all the values that make $- 2 (x^{2} + 3) (2 x - 4) {(3 x + 1)}^{2} = 0$ true.

$x = i \sqrt{3}, - i \sqrt{3}, 2, - \frac{1}{3}$

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (- \frac{1}{3}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 2 ({(0)}^{2} + 3) (2 (0) - 4) {(3 (0) + 1)}^{2}$

Solve the equation.

Remove parentheses.

$y = - 2 (0^{2} + 3) (2 (0) - 4) {(3 (0) + 1)}^{2}$

Remove parentheses.

$y = - 2 ({(0)}^{2} + 3) (2 (0) - 4) {(3 (0) + 1)}^{2}$

Simplify $- 2 ({(0)}^{2} + 3) (2 (0) - 4) {(3 (0) + 1)}^{2}$ .

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

$y = - 2 (0 + 3) (2 (0) - 4) {(3 (0) + 1)}^{2}$

Add $0$ and $3$ .

$y = - 2 \cdot 3 (2 (0) - 4) {(3 (0) + 1)}^{2}$

Multiply $- 2$ by $3$ .

$y = - 6 (2 (0) - 4) {(3 (0) + 1)}^{2}$

Multiply $2$ by $0$ .

$y = - 6 (0 - 4) {(3 (0) + 1)}^{2}$

Subtract $4$ from $0$ .

$y = - 6 \cdot - 4 {(3 (0) + 1)}^{2}$

Multiply $- 6$ by $- 4$ .

$y = 24 {(3 (0) + 1)}^{2}$

Multiply $3$ by $0$ .

$y = 24 {(0 + 1)}^{2}$

Add $0$ and $1$ .

$y = 24 \cdot 1^{2}$

One to any power is one.

$y = 24 \cdot 1$

Multiply $24$ by $1$ .

$y = 24$

y-intercept(s) in point form.

y-intercept(s): $(0, 24)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (- \frac{1}{3}, 0)$

y-intercept(s): $(0, 24)$

Step 4

Find the x and y Intercepts f(x)=-2(x^2+3)(2x-4)(3x+1)^2

Related Questions

Support LCMGCF.com by disabling your adblocker.