Algebra Examples

$9 x^{2} - 1 - {(3 x - 2)}^{2} = 0$

Step 1

Rewrite ${(3 x - 2)}^{2}$ as $(3 x - 2) (3 x - 2)$ .

$9 x^{2} - 1 - ((3 x - 2) (3 x - 2)) = 0$

Step 2

Expand $(3 x - 2) (3 x - 2)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$9 x^{2} - 1 - (3 x (3 x - 2) - 2 (3 x - 2)) = 0$

Step 2.2

Apply the distributive property.

$9 x^{2} - 1 - (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x - 2)) = 0$

Step 2.3

Apply the distributive property.

$9 x^{2} - 1 - (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$9 x^{2} - 1 - (3 \cdot (3 x \cdot x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Step 3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1

Move $x$ .

$9 x^{2} - 1 - (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Step 3.1.2.2

Multiply $x$ by $x$ .

$9 x^{2} - 1 - (3 \cdot (3 x^{2}) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Step 3.1.3

Multiply $3$ by $3$ .

$9 x^{2} - 1 - (9 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Step 3.1.4

Multiply $- 2$ by $3$ .

$9 x^{2} - 1 - (9 x^{2} - 6 x - 2 (3 x) - 2 \cdot - 2) = 0$

Step 3.1.5

Multiply $3$ by $- 2$ .

$9 x^{2} - 1 - (9 x^{2} - 6 x - 6 x - 2 \cdot - 2) = 0$

Step 3.1.6

Multiply $- 2$ by $- 2$ .

$9 x^{2} - 1 - (9 x^{2} - 6 x - 6 x + 4) = 0$

Step 3.2

Subtract $6 x$ from $- 6 x$ .

$9 x^{2} - 1 - (9 x^{2} - 12 x + 4) = 0$

Step 4

Apply the distributive property.

$9 x^{2} - 1 - (9 x^{2}) - (- 12 x) - 1 \cdot 4 = 0$

Step 5

Simplify.

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Step 5.1

Multiply $9$ by $- 1$ .

$9 x^{2} - 1 - 9 x^{2} - (- 12 x) - 1 \cdot 4 = 0$

Step 5.2

Multiply $- 12$ by $- 1$ .

$9 x^{2} - 1 - 9 x^{2} + 12 x - 1 \cdot 4 = 0$

Step 5.3

Multiply $- 1$ by $4$ .

$9 x^{2} - 1 - 9 x^{2} + 12 x - 4 = 0$

Step 6

Subtract $9 x^{2}$ from $9 x^{2}$ .

$0 - 1 + 12 x - 4 = 0$

Step 7

Subtract $1$ from $0$ .

$- 1 + 12 x - 4 = 0$

Step 8

Subtract $4$ from $- 1$ .

$12 x - 5 = 0$

Step 9

Add $5$ to both sides of the equation.

$12 x = 5$

Step 10

Divide each term in $12 x = 5$ by $12$ and simplify.

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Step 10.1

Divide each term in $12 x = 5$ by $12$ .

$\frac{12 x}{12} = \frac{5}{12}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $12$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{5}{12}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{12}$