Algebra Examples

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

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

Step 1

Set

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

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

0

$0$ .

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

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

Step 2

Solve for

x

$x$ .

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Step 2.1

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x^{2} = 0

$x^{2} = 0$

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

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

4 x + 3 = 0

$4 x + 3 = 0$

Step 2.2

Set

x^{2}

$x^{2}$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.2.1

Set

x^{2}

$x^{2}$ equal to

0

$0$ .

x^{2} = 0

$x^{2} = 0$

Step 2.2.2

Solve

x^{2} = 0

$x^{2} = 0$ for

x

$x$ .

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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}

$x = \pm \sqrt{0}$

Step 2.2.2.2

Simplify

\pm \sqrt{0}

$\pm \sqrt{0}$ .

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Step 2.2.2.2.1

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

x = \pm \sqrt{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

$x = \pm 0$

Step 2.2.2.2.3

Plus or minus

0

$0$ is

0

$0$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 2.3

Set

{(2 x - 1)}^{3}

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

0

$0$ and solve for

x

$x$ .

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Step 2.3.1

Set

{(2 x - 1)}^{3}

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

0

$0$ .

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

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

Step 2.3.2

Solve

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

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

x

$x$ .

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Step 2.3.2.1

Set the

2 x - 1

$2 x - 1$ equal to

0

$0$ .

2 x - 1 = 0

$2 x - 1 = 0$

Step 2.3.2.2

Solve for

x

$x$ .

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Step 2.3.2.2.1

Add

1

$1$ to both sides of the equation.

2 x = 1

$2 x = 1$

Step 2.3.2.2.2

Divide each term in

2 x = 1

$2 x = 1$ by

2

$2$ and simplify.

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Step 2.3.2.2.2.1

Divide each term in

2 x = 1

$2 x = 1$ by

2

$2$ .

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

$2$ .

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Step 2.3.2.2.2.2.1.1

Cancel the common factor.

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

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

Step 2.3.2.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{1}{2}

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

x = \frac{1}{2}

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

x = \frac{1}{2}

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

x = \frac{1}{2}

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

x = \frac{1}{2}

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

x = \frac{1}{2}

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

x = \frac{1}{2}

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

Step 2.4

Set

4 x + 3

$4 x + 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.4.1

Set

4 x + 3

$4 x + 3$ equal to

0

$0$ .

4 x + 3 = 0

$4 x + 3 = 0$

Step 2.4.2

Solve

4 x + 3 = 0

$4 x + 3 = 0$ for

x

$x$ .

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Step 2.4.2.1

Subtract

3

$3$ from both sides of the equation.

4 x = - 3

$4 x = - 3$

Step 2.4.2.2

Divide each term in

4 x = - 3

$4 x = - 3$ by

4

$4$ and simplify.

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Step 2.4.2.2.1

Divide each term in

4 x = - 3

$4 x = - 3$ by

4

$4$ .

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

$4$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

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

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

Step 2.4.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 3}{4}

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

x = \frac{- 3}{4}

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

x = \frac{- 3}{4}

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

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}

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

x = - \frac{3}{4}

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

x = - \frac{3}{4}

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

x = - \frac{3}{4}

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

x = - \frac{3}{4}

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

Step 2.5

The final solution is all the values that make

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

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

x = 0, \frac{1}{2}, - \frac{3}{4}

$x = 0, \frac{1}{2}, - \frac{3}{4}$

x = 0, \frac{1}{2}, - \frac{3}{4}

$x = 0, \frac{1}{2}, - \frac{3}{4}$

Step 3