Algebra Examples

$f (x) = 2 x + x^{2}$

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 $x$ .

$0 = 2 x + x^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x + x^{2} = 0$ .

$2 x + x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2 u + u^{2} = 0$

Step 1.2.2.2

Factor $u$ out of $2 u + u^{2}$ .

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Step 1.2.2.2.1

Factor $u$ out of $2 u$ .

$u \cdot 2 + u^{2} = 0$

Step 1.2.2.2.2

Factor $u$ out of $u^{2}$ .

$u \cdot 2 + u \cdot u = 0$

Step 1.2.2.2.3

Factor $u$ out of $u \cdot 2 + u \cdot u$ .

$u (2 + u) = 0$

Step 1.2.2.3

Replace all occurrences of $u$ with $x$ .

$x (2 + x) = 0$

Step 1.2.3

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

$x = 0$

$2 + x = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $2 + x$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 + x$ equal to $0$ .

$2 + x = 0$

Step 1.2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.6

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

$x = 0, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 2, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $2 (0) + {(0)}^{2}$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $2$ by $0$ .

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

Step 2.2.3.1.2

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

$y = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 2, 0)$

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

Step 4