Algebra Examples

$y = x \sqrt{x + 2}$

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 $x$ .

$0 = x \sqrt{x + 2}$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$x \sqrt{x + 2} = 0$

Step 1.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(x \sqrt{x + 2})}^{2} = 0^{2}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 2}$ as ${(x + 2)}^{\frac{1}{2}}$ .

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${(x {(x + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1

Apply the product rule to $x {(x + 2)}^{\frac{1}{2}}$ .

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

Step 1.2.3.2.1.2

Multiply the exponents in ${({(x + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.3.2.1.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.2.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.2.2.2

Rewrite the expression.

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

Step 1.2.3.2.1.3

Simplify.

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

Step 1.2.3.2.1.4

Apply the distributive property.

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

Step 1.2.3.2.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.3.2.1.5.1

Multiply $x^{2}$ by $x$ .

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Step 1.2.3.2.1.5.1.1

Raise $x$ to the power of $1$ .

$x^{2} x^{1} + x^{2} \cdot 2 = 0^{2}$

Step 1.2.3.2.1.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} \cdot 2 = 0^{2}$

Step 1.2.3.2.1.5.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot 2 = 0^{2}$

Step 1.2.3.2.1.6

Move $2$ to the left of $x^{2}$ .

$x^{3} + 2 x^{2} = 0^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

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

$x^{3} + 2 x^{2} = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

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

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Step 1.2.4.1.1

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

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

Step 1.2.4.1.2

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

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

Step 1.2.4.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 2$ .

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

Step 1.2.4.2

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

$x^{2} = 0$

$x + 2 = 0$

Step 1.2.4.3

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

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Step 1.2.4.3.1

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

$x^{2} = 0$

Step 1.2.4.3.2

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

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Step 1.2.4.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.3.2.2.1

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

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

Step 1.2.4.3.2.2.2

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

$x = \pm 0$

Step 1.2.4.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4.4

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

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Step 1.2.4.4.1

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

$x + 2 = 0$

Step 1.2.4.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.4.5

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

$y = (0) \sqrt{(0) + 2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 \sqrt{0 + 2}$

Step 2.2.2

Remove parentheses.

$y = (0) \sqrt{(0) + 2}$

Step 2.2.3

Simplify $(0) \sqrt{(0) + 2}$ .

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Step 2.2.3.1

Add $0$ and $2$ .

$y = 0 \sqrt{2}$

Step 2.2.3.2

Multiply $0$ by $\sqrt{2}$ .

$y = 0$

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