Algebra Examples

$9 x (x + 6) - {(3 x + 1)}^{2} = 1$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

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

Step 1.2

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

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Step 1.2.1

Move $x$ .

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

Step 1.2.2

Multiply $x$ by $x$ .

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

Step 1.3

Multiply $6$ by $9$ .

$9 x^{2} + 54 x - {(3 x + 1)}^{2} = 1$

Step 2

Subtract $1$ from both sides of the equation.

$9 x^{2} + 54 x - {(3 x + 1)}^{2} - 1 = 0$

Step 3

Simplify $9 x^{2} + 54 x - {(3 x + 1)}^{2} - 1$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

$9 x^{2} + 54 x - ((3 x + 1) (3 x + 1)) - 1 = 0$

Step 3.1.2

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$9 x^{2} + 54 x - (3 x (3 x + 1) + 1 (3 x + 1)) - 1 = 0$

Step 3.1.2.2

Apply the distributive property.

$9 x^{2} + 54 x - (3 x (3 x) + 3 x \cdot 1 + 1 (3 x + 1)) - 1 = 0$

Step 3.1.2.3

Apply the distributive property.

$9 x^{2} + 54 x - (3 x (3 x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 1 = 0$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Rewrite using the commutative property of multiplication.

$9 x^{2} + 54 x - (3 \cdot 3 x \cdot x + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 1 = 0$

Step 3.1.3.1.2

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

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Step 3.1.3.1.2.1

Move $x$ .

$9 x^{2} + 54 x - (3 \cdot 3 (x \cdot x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 1 = 0$

Step 3.1.3.1.2.2

Multiply $x$ by $x$ .

$9 x^{2} + 54 x - (3 \cdot 3 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 1 = 0$

Step 3.1.3.1.3

Multiply $3$ by $3$ .

$9 x^{2} + 54 x - (9 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 1 = 0$

Step 3.1.3.1.4

Multiply $3$ by $1$ .

$9 x^{2} + 54 x - (9 x^{2} + 3 x + 1 (3 x) + 1 \cdot 1) - 1 = 0$

Step 3.1.3.1.5

Multiply $3 x$ by $1$ .

$9 x^{2} + 54 x - (9 x^{2} + 3 x + 3 x + 1 \cdot 1) - 1 = 0$

Step 3.1.3.1.6

Multiply $1$ by $1$ .

$9 x^{2} + 54 x - (9 x^{2} + 3 x + 3 x + 1) - 1 = 0$

Step 3.1.3.2

Add $3 x$ and $3 x$ .

$9 x^{2} + 54 x - (9 x^{2} + 6 x + 1) - 1 = 0$

Step 3.1.4

Apply the distributive property.

$9 x^{2} + 54 x - (9 x^{2}) - (6 x) - 1 \cdot 1 - 1 = 0$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $9$ by $- 1$ .

$9 x^{2} + 54 x - 9 x^{2} - (6 x) - 1 \cdot 1 - 1 = 0$

Step 3.1.5.2

Multiply $6$ by $- 1$ .

$9 x^{2} + 54 x - 9 x^{2} - 6 x - 1 \cdot 1 - 1 = 0$

Step 3.1.5.3

Multiply $- 1$ by $1$ .

$9 x^{2} + 54 x - 9 x^{2} - 6 x - 1 - 1 = 0$

Step 3.2

Combine the opposite terms in $9 x^{2} + 54 x - 9 x^{2} - 6 x - 1 - 1$ .

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Step 3.2.1

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

$54 x + 0 - 6 x - 1 - 1 = 0$

Step 3.2.2

Add $54 x$ and $0$ .

$54 x - 6 x - 1 - 1 = 0$

Step 3.3

Subtract $6 x$ from $54 x$ .

$48 x - 1 - 1 = 0$

Step 3.4

Subtract $1$ from $- 1$ .

$48 x - 2 = 0$

Step 4

Factor $2$ out of $48 x - 2$ .

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Step 4.1

Factor $2$ out of $48 x$ .

$2 (24 x) - 2 = 0$

Step 4.2

Factor $2$ out of $- 2$ .

$2 (24 x) + 2 (- 1) = 0$

Step 4.3

Factor $2$ out of $2 (24 x) + 2 (- 1)$ .

$2 (24 x - 1) = 0$

Step 5

Divide each term in $2 (24 x - 1) = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 (24 x - 1) = 0$ by $2$ .

$\frac{2 (24 x - 1)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 (24 x - 1)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $24 x - 1$ by $1$ .

$24 x - 1 = \frac{0}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$24 x - 1 = 0$

Step 6

Add $1$ to both sides of the equation.

$24 x = 1$

Step 7

Divide each term in $24 x = 1$ by $24$ and simplify.

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Step 7.1

Divide each term in $24 x = 1$ by $24$ .

$\frac{24 x}{24} = \frac{1}{24}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $24$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{1}{24}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{24}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{24}$

Decimal Form:

$x = 0.041\overline{6}$