Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

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 $0$ .

$x = 0$

${(x - 2)}^{2} = 0$

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

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

${(x - 2)}^{2} = 0$

Step 1.2.4.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 1.2.4.2.1

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

$x - 2 = 0$

Step 1.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

Set ${(4 x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set ${(4 x - 2)}^{2}$ equal to $0$ .

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

Step 1.2.5.2

Solve ${(4 x - 2)}^{2} = 0$ for $x$ .

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Step 1.2.5.2.1

Factor the left side of the equation.

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Step 1.2.5.2.1.1

Factor $2$ out of $4 x - 2$ .

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Step 1.2.5.2.1.1.1

Factor $2$ out of $4 x$ .

${(2 (2 x) - 2)}^{2} = 0$

Step 1.2.5.2.1.1.2

Factor $2$ out of $- 2$ .

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

Step 1.2.5.2.1.1.3

Factor $2$ out of $2 (2 x) + 2 (- 1)$ .

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

Step 1.2.5.2.1.2

Apply the product rule to $2 (2 x - 1)$ .

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

Step 1.2.5.2.2

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

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Step 1.2.5.2.2.1

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

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

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $2^{2}$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.2.2.1.2

Divide ${(2 x - 1)}^{2}$ by $1$ .

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

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

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

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

Step 1.2.5.2.2.3.2

Divide $0$ by $4$ .

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

Step 1.2.5.2.3

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 1.2.5.2.4

Solve for $x$ .

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Step 1.2.5.2.4.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 1.2.5.2.4.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 1.2.5.2.4.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.2.5.2.4.2.2

Simplify the left side.

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Step 1.2.5.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.2.5.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 1.2.6

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

$x = 0, 2, \frac{1}{2}$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Multiply $2$ by $0$ .

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

Step 2.2.3.2

Subtract $2$ from $0$ .

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Multiply $0$ by $4$ .

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

Step 2.2.3.5

Multiply $4$ by $0$ .

$y = 0 {(0 - 2)}^{2}$

Step 2.2.3.6

Subtract $2$ from $0$ .

$y = 0 {(- 2)}^{2}$

Step 2.2.3.7

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

$y = 0 \cdot 4$

Step 2.2.3.8

Multiply $0$ by $4$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4