Pre-Algebra Examples

$(3 x - 11) {(2 x + 9)}^{2} x = 180$

Step 1

Simplify $(3 x - 11) {(2 x + 9)}^{2} x$ .

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Step 1.1

Rewrite ${(2 x + 9)}^{2}$ as $(2 x + 9) (2 x + 9)$ .

$(3 x - 11) ((2 x + 9) (2 x + 9)) x = 180$

Step 1.2

Expand $(2 x + 9) (2 x + 9)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$(3 x - 11) (2 x (2 x + 9) + 9 (2 x + 9)) x = 180$

Step 1.2.2

Apply the distributive property.

$(3 x - 11) (2 x (2 x) + 2 x \cdot 9 + 9 (2 x + 9)) x = 180$

Step 1.2.3

Apply the distributive property.

$(3 x - 11) (2 x (2 x) + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9) x = 180$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$(3 x - 11) (2 \cdot 2 x \cdot x + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9) x = 180$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

$(3 x - 11) (2 \cdot 2 (x \cdot x) + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9) x = 180$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$(3 x - 11) (2 \cdot 2 x^{2} + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9) x = 180$

Step 1.3.1.3

Multiply $2$ by $2$ .

$(3 x - 11) (4 x^{2} + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9) x = 180$

Step 1.3.1.4

Multiply $9$ by $2$ .

$(3 x - 11) (4 x^{2} + 18 x + 9 (2 x) + 9 \cdot 9) x = 180$

Step 1.3.1.5

Multiply $2$ by $9$ .

$(3 x - 11) (4 x^{2} + 18 x + 18 x + 9 \cdot 9) x = 180$

Step 1.3.1.6

Multiply $9$ by $9$ .

$(3 x - 11) (4 x^{2} + 18 x + 18 x + 81) x = 180$

Step 1.3.2

Add $18 x$ and $18 x$ .

$(3 x - 11) (4 x^{2} + 36 x + 81) x = 180$

Step 1.4

Expand $(3 x - 11) (4 x^{2} + 36 x + 81)$ by multiplying each term in the first expression by each term in the second expression.

$(3 x (4 x^{2}) + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Rewrite using the commutative property of multiplication.

$(3 \cdot 4 x \cdot x^{2} + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.5.1.2.1

Move $x^{2}$ .

$(3 \cdot 4 (x^{2} x) + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.5.1.2.2.1

Raise $x$ to the power of $1$ .

$(3 \cdot 4 (x^{2} x^{1}) + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(3 \cdot 4 x^{2 + 1} + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.2.3

Add $2$ and $1$ .

$(3 \cdot 4 x^{3} + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.3

Multiply $3$ by $4$ .

$(12 x^{3} + 3 x (36 x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.4

Rewrite using the commutative property of multiplication.

$(12 x^{3} + 3 \cdot 36 x \cdot x + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.5

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

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Step 1.5.1.5.1

Move $x$ .

$(12 x^{3} + 3 \cdot 36 (x \cdot x) + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.5.2

Multiply $x$ by $x$ .

$(12 x^{3} + 3 \cdot 36 x^{2} + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.6

Multiply $3$ by $36$ .

$(12 x^{3} + 108 x^{2} + 3 x \cdot 81 - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.7

Multiply $81$ by $3$ .

$(12 x^{3} + 108 x^{2} + 243 x - 11 (4 x^{2}) - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.8

Multiply $4$ by $- 11$ .

$(12 x^{3} + 108 x^{2} + 243 x - 44 x^{2} - 11 (36 x) - 11 \cdot 81) x = 180$

Step 1.5.1.9

Multiply $36$ by $- 11$ .

$(12 x^{3} + 108 x^{2} + 243 x - 44 x^{2} - 396 x - 11 \cdot 81) x = 180$

Step 1.5.1.10

Multiply $- 11$ by $81$ .

$(12 x^{3} + 108 x^{2} + 243 x - 44 x^{2} - 396 x - 891) x = 180$

Step 1.5.2

Simplify terms.

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Step 1.5.2.1

Subtract $44 x^{2}$ from $108 x^{2}$ .

$(12 x^{3} + 64 x^{2} + 243 x - 396 x - 891) x = 180$

Step 1.5.2.2

Subtract $396 x$ from $243 x$ .

$(12 x^{3} + 64 x^{2} - 153 x - 891) x = 180$

Step 1.5.2.3

Apply the distributive property.

$12 x^{3} x + 64 x^{2} x - 153 x \cdot x - 891 x = 180$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.6.1.1

Move $x$ .

$12 (x \cdot x^{3}) + 64 x^{2} x - 153 x \cdot x - 891 x = 180$

Step 1.6.1.2

Multiply $x$ by $x^{3}$ .

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Step 1.6.1.2.1

Raise $x$ to the power of $1$ .

$12 (x^{1} x^{3}) + 64 x^{2} x - 153 x \cdot x - 891 x = 180$

Step 1.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{1 + 3} + 64 x^{2} x - 153 x \cdot x - 891 x = 180$

Step 1.6.1.3

Add $1$ and $3$ .

$12 x^{4} + 64 x^{2} x - 153 x \cdot x - 891 x = 180$

Step 1.6.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.6.2.1

Move $x$ .

$12 x^{4} + 64 (x \cdot x^{2}) - 153 x \cdot x - 891 x = 180$

Step 1.6.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.6.2.2.1

Raise $x$ to the power of $1$ .

$12 x^{4} + 64 (x^{1} x^{2}) - 153 x \cdot x - 891 x = 180$

Step 1.6.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{4} + 64 x^{1 + 2} - 153 x \cdot x - 891 x = 180$

Step 1.6.2.3

Add $1$ and $2$ .

$12 x^{4} + 64 x^{3} - 153 x \cdot x - 891 x = 180$

Step 1.6.3

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

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Step 1.6.3.1

Move $x$ .

$12 x^{4} + 64 x^{3} - 153 (x \cdot x) - 891 x = 180$

Step 1.6.3.2

Multiply $x$ by $x$ .

$12 x^{4} + 64 x^{3} - 153 x^{2} - 891 x = 180$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 5.08093716, - 3.76829935, - 0.21025244, 3.72615563$

Step 3