Algebra Examples

$81 - 4 x^{2} = 0$

Step 1

Rewrite $81$ as $9^{2}$ .

$9^{2} - 4 x^{2} = 0$

Step 2

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

$9^{2} - {(2 x)}^{2} = 0$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = 2 x$ .

$(9 + 2 x) (9 - (2 x)) = 0$

Step 4

Multiply $2$ by $- 1$ .

$(9 + 2 x) (9 - 2 x) = 0$

Step 5

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

$9 + 2 x = 0$

$9 - 2 x = 0$

Step 6

Set $9 + 2 x$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $9 + 2 x$ equal to $0$ .

$9 + 2 x = 0$

Step 6.2

Solve $9 + 2 x = 0$ for $x$ .

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Step 6.2.1

Subtract $9$ from both sides of the equation.

$2 x = - 9$

Step 6.2.2

Divide each term in $2 x = - 9$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = - 9$ by $2$ .

$\frac{2 x}{2} = \frac{- 9}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 9}{2}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{2}$

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.

$x = - \frac{9}{2}$

Step 7

Set $9 - 2 x$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $9 - 2 x$ equal to $0$ .

$9 - 2 x = 0$

Step 7.2

Solve $9 - 2 x = 0$ for $x$ .

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Step 7.2.1

Subtract $9$ from both sides of the equation.

$- 2 x = - 9$

Step 7.2.2

Divide each term in $- 2 x = - 9$ by $- 2$ and simplify.

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Step 7.2.2.1

Divide each term in $- 2 x = - 9$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 9}{- 2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 9}{- 2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{- 2}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{9}{2}$

Step 8

The final solution is all the values that make $(9 + 2 x) (9 - 2 x) = 0$ true.

$x = - \frac{9}{2}, \frac{9}{2}$