Linear Algebra Examples

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

Step 1

Subtract $b x$ from both sides of the equation.

$a x^{2} = - b x$

Step 2

Divide each term in $a x^{2} = - b x$ by $x^{2}$ and simplify.

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Step 2.1

Divide each term in $a x^{2} = - b x$ by $x^{2}$ .

$\frac{a x^{2}}{x^{2}} = \frac{- b x}{x^{2}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$\frac{a x^{2}}{x^{2}} = \frac{- b x}{x^{2}}$

Step 2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- b x}{x^{2}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Factor $x$ out of $- b x$ .

$a = \frac{x (- b)}{x^{2}}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

$a = \frac{x (- b)}{x \cdot x}$

Step 2.3.1.2.2

Cancel the common factor.

$a = \frac{x (- b)}{x \cdot x}$

Step 2.3.1.2.3

Rewrite the expression.

$a = \frac{- b}{x}$

Step 2.3.2

Move the negative in front of the fraction.

$a = - \frac{b}{x}$

Step 3

Set the denominator in $\frac{b}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

The domain is all values of $b$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${b | b \neq 0}$