Algebra Examples

$f (x) = - 2 x^{2} {(2 x - 1)}^{3} (4 x + 3)$

Step 1

Set $- 2 x^{2} {(2 x - 1)}^{3} (4 x + 3)$ equal to $0$ .

$- 2 x^{2} {(2 x - 1)}^{3} (4 x + 3) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x^{2} = 0$

${(2 x - 1)}^{3} = 0$

$4 x + 3 = 0$

Step 2.2

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

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Step 2.2.1

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

$x^{2} = 0$

Step 2.2.2

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

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Step 2.2.2.1

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

$x = \pm \sqrt{0}$

Step 2.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

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

$x = \pm 0$

Step 2.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3

Set ${(2 x - 1)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set ${(2 x - 1)}^{3}$ equal to $0$ .

${(2 x - 1)}^{3} = 0$

Step 2.3.2

Solve ${(2 x - 1)}^{3} = 0$ for $x$ .

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Step 2.3.2.1

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 2.3.2.2

Solve for $x$ .

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Step 2.3.2.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.3.2.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 2.3.2.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 2.3.2.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 2.3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 2.4

Set $4 x + 3$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Step 2.4.2

Solve $4 x + 3 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Step 2.4.2.2

Divide each term in $4 x = - 3$ by $4$ and simplify.

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Step 2.4.2.2.1

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 2.5

The final solution is all the values that make $- 2 x^{2} {(2 x - 1)}^{3} (4 x + 3) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $2$ )

$x = \frac{1}{2}$ (Multiplicity of $3$ )

$x = - \frac{3}{4}$ (Multiplicity of $1$ )

$x = 0$ (Multiplicity of $2$ )

$x = \frac{1}{2}$ (Multiplicity of $3$ )

$x = - \frac{3}{4}$ (Multiplicity of $1$ )

Step 3