Algebra Examples

$2 x^{3} - 2 x^{2} - 84 x$

Step 1

Write $2 x^{3} - 2 x^{2} - 84 x$ as an equation.

$y = 2 x^{3} - 2 x^{2} - 84 x$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = 2 x^{3} - 2 x^{2} - 84 x$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $2 x^{3} - 2 x^{2} - 84 x = 0$ .

$2 x^{3} - 2 x^{2} - 84 x = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $2 x$ out of $2 x^{3} - 2 x^{2} - 84 x$ .

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Step 2.2.2.1.1

Factor $2 x$ out of $2 x^{3}$ .

$2 x (x^{2}) - 2 x^{2} - 84 x = 0$

Step 2.2.2.1.2

Factor $2 x$ out of $- 2 x^{2}$ .

$2 x (x^{2}) + 2 x (- x) - 84 x = 0$

Step 2.2.2.1.3

Factor $2 x$ out of $- 84 x$ .

$2 x (x^{2}) + 2 x (- x) + 2 x (- 42) = 0$

Step 2.2.2.1.4

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- x)$ .

$2 x (x^{2} - x) + 2 x (- 42) = 0$

Step 2.2.2.1.5

Factor $2 x$ out of $2 x (x^{2} - x) + 2 x (- 42)$ .

$2 x (x^{2} - x - 42) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor $x^{2} - x - 42$ using the AC method.

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Step 2.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$2 x ((x - 7) (x + 6)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$2 x (x - 7) (x + 6) = 0$

Step 2.2.3

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

$x = 0$

$x - 7 = 0$

$x + 6 = 0$

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 2.2.5.2

Add $7$ to both sides of the equation.

$x = 7$

Step 2.2.6

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.2.6.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.2.7

The final solution is all the values that make $2 x (x - 7) (x + 6) = 0$ true.

$x = 0, 7, - 6$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (7, 0), (- 6, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = 2 {(0)}^{3} - 2 {(0)}^{2} - 84 \cdot 0$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 2 \cdot 0^{3} - 2 {(0)}^{2} - 84 \cdot 0$

Step 3.2.2

Remove parentheses.

$y = 2 \cdot 0^{3} - 2 \cdot 0^{2} - 84 \cdot 0$

Step 3.2.3

Remove parentheses.

$y = 2 {(0)}^{3} - 2 {(0)}^{2} - 84 \cdot 0$

Step 3.2.4

Simplify $2 {(0)}^{3} - 2 {(0)}^{2} - 84 \cdot 0$ .

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

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

$y = 2 \cdot 0 - 2 {(0)}^{2} - 84 \cdot 0$

Step 3.2.4.1.2

Multiply $2$ by $0$ .

$y = 0 - 2 {(0)}^{2} - 84 \cdot 0$

Step 3.2.4.1.3

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

$y = 0 - 2 \cdot 0 - 84 \cdot 0$

Step 3.2.4.1.4

Multiply $- 2$ by $0$ .

$y = 0 + 0 - 84 \cdot 0$

Step 3.2.4.1.5

Multiply $- 84$ by $0$ .

$y = 0 + 0 + 0$

Step 3.2.4.2

Simplify by adding numbers.

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Step 3.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (7, 0), (- 6, 0)$

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

Step 5