Algebra Examples

$f (x) = 5 x^{2} (x^{2} - 7) {(x - 1)}^{2}$

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 $x$ .

$0 = 5 x^{2} (x^{2} - 7) {(x - 1)}^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5 x^{2} (x^{2} - 7) {(x - 1)}^{2} = 0$ .

$5 x^{2} (x^{2} - 7) {(x - 1)}^{2} = 0$

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 $0$ .

$x^{2} = 0$

$x^{2} - 7 = 0$

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

Step 1.2.3

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

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Step 1.2.3.1

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

$x^{2} = 0$

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Step 1.2.4.2

Solve $x^{2} - 7 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $7$ to both sides of the equation.

$x^{2} = 7$

Step 1.2.4.2.2

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

$x = \pm \sqrt{7}$

Step 1.2.4.2.3

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

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Step 1.2.4.2.3.1

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

$x = \sqrt{7}$

Step 1.2.4.2.3.2

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

$x = - \sqrt{7}$

Step 1.2.4.2.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

$x - 1 = 0$

Step 1.2.5.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6

The final solution is all the values that make $5 x^{2} (x^{2} - 7) {(x - 1)}^{2} = 0$ true.

$x = 0, \sqrt{7}, - \sqrt{7}, 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\sqrt{7}, 0), (- \sqrt{7}, 0), (1, 0)$

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 $y$ .

$y = 5 {(0)}^{2} ({(0)}^{2} - 7) {((0) - 1)}^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 5 \cdot (0^{2} ({(0)}^{2} - 7) {((0) - 1)}^{2})$

Step 2.2.2

Remove parentheses.

$y = 5 \cdot (0^{2} (0^{2} - 7) {((0) - 1)}^{2})$

Step 2.2.3

Remove parentheses.

$y = 5 \cdot (0^{2} (0^{2} - 7) {(0 - 1)}^{2})$

Step 2.2.4

Remove parentheses.

$y = 5 {(0)}^{2} ({(0)}^{2} - 7) {((0) - 1)}^{2}$

Step 2.2.5

Simplify $5 {(0)}^{2} ({(0)}^{2} - 7) {((0) - 1)}^{2}$ .

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Step 2.2.5.1

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

$y = 5 \cdot 0 ({(0)}^{2} - 7) {((0) - 1)}^{2}$

Step 2.2.5.2

Multiply $5$ by $0$ .

$y = 0 ({(0)}^{2} - 7) {((0) - 1)}^{2}$

Step 2.2.5.3

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

$y = 0 (0 - 7) {((0) - 1)}^{2}$

Step 2.2.5.4

Subtract $7$ from $0$ .

$y = 0 \cdot - 7 {((0) - 1)}^{2}$

Step 2.2.5.5

Multiply $0$ by $- 7$ .

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

Step 2.2.5.6

Subtract $1$ from $0$ .

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

Step 2.2.5.7

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

$y = 0 \cdot 1$

Step 2.2.5.8

Multiply $0$ by $1$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\sqrt{7}, 0), (- \sqrt{7}, 0), (1, 0)$

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

Step 4