Algebra Examples

$32 x^{2} - 2 = 0$

Step 1

Factor $2$ out of $32 x^{2} - 2$ .

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Step 1.1

Factor $2$ out of $32 x^{2}$ .

$2 (16 x^{2}) - 2 = 0$

Step 1.2

Factor $2$ out of $- 2$ .

$2 (16 x^{2}) + 2 (- 1) = 0$

Step 1.3

Factor $2$ out of $2 (16 x^{2}) + 2 (- 1)$ .

$2 (16 x^{2} - 1) = 0$

Step 2

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

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

Step 3

Rewrite $1$ as $1^{2}$ .

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

Step 4

Factor.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = 1$ .

$2 ((4 x + 1) (4 x - 1)) = 0$

Step 4.2

Remove unnecessary parentheses.

$2 (4 x + 1) (4 x - 1) = 0$

Step 5

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

$4 x + 1 = 0$

$4 x - 1 = 0$

Step 6

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 6.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 6.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 6.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 7

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 7.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 7.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 8

The final solution is all the values that make $2 (4 x + 1) (4 x - 1) = 0$ true.

$x = - \frac{1}{4}, \frac{1}{4}$