Pre-Algebra Examples

$3 (x - 6) (2.2 x - 6) = 800$

Step 1

Divide each term in $3 (x - 6) (2.2 x - 6) = 800$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 (x - 6) (2.2 x - 6) = 800$ by $3$ .

$\frac{3 (x - 6) (2.2 x - 6)}{3} = \frac{800}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 (x - 6) (2.2 x - 6)}{3} = \frac{800}{3}$

Step 1.2.1.2

Divide $(x - 6) (2.2 x - 6)$ by $1$ .

$(x - 6) (2.2 x - 6) = \frac{800}{3}$

Step 1.2.2

Expand $(x - 6) (2.2 x - 6)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$x (2.2 x - 6) - 6 (2.2 x - 6) = \frac{800}{3}$

Step 1.2.2.2

Apply the distributive property.

$x (2.2 x) + x \cdot - 6 - 6 (2.2 x - 6) = \frac{800}{3}$

Step 1.2.2.3

Apply the distributive property.

$x (2.2 x) + x \cdot - 6 - 6 (2.2 x) - 6 \cdot - 6 = \frac{800}{3}$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Rewrite using the commutative property of multiplication.

$2.2 x \cdot x + x \cdot - 6 - 6 (2.2 x) - 6 \cdot - 6 = \frac{800}{3}$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

Move $x$ .

$2.2 (x \cdot x) + x \cdot - 6 - 6 (2.2 x) - 6 \cdot - 6 = \frac{800}{3}$

Step 1.2.3.1.2.2

Multiply $x$ by $x$ .

$2.2 x^{2} + x \cdot - 6 - 6 (2.2 x) - 6 \cdot - 6 = \frac{800}{3}$

Step 1.2.3.1.3

Move $- 6$ to the left of $x$ .

$2.2 x^{2} - 6 \cdot x - 6 (2.2 x) - 6 \cdot - 6 = \frac{800}{3}$

Step 1.2.3.1.4

Multiply $2.2$ by $- 6$ .

$2.2 x^{2} - 6 x - 13.2 x - 6 \cdot - 6 = \frac{800}{3}$

Step 1.2.3.1.5

Multiply $- 6$ by $- 6$ .

$2.2 x^{2} - 6 x - 13.2 x + 36 = \frac{800}{3}$

Step 1.2.3.2

Subtract $13.2 x$ from $- 6 x$ .

$2.2 x^{2} - 19.2 x + 36 = \frac{800}{3}$

Step 2

Subtract $\frac{800}{3}$ from both sides of the equation.

$2.2 x^{2} - 19.2 x + 36 - \frac{800}{3} = 0$

Step 3

Simplify $2.2 x^{2} - 19.2 x + 36 - \frac{800}{3}$ .

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Step 3.1

To write $36$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2.2 x^{2} - 19.2 x + 36 \cdot \frac{3}{3} - \frac{800}{3} = 0$

Step 3.2

Combine $36$ and $\frac{3}{3}$ .

$2.2 x^{2} - 19.2 x + \frac{36 \cdot 3}{3} - \frac{800}{3} = 0$

Step 3.3

Combine the numerators over the common denominator.

$2.2 x^{2} - 19.2 x + \frac{36 \cdot 3 - 800}{3} = 0$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $36$ by $3$ .

$2.2 x^{2} - 19.2 x + \frac{108 - 800}{3} = 0$

Step 3.4.2

Subtract $800$ from $108$ .

$2.2 x^{2} - 19.2 x + \frac{- 692}{3} = 0$

Step 3.5

Move the negative in front of the fraction.

$2.2 x^{2} - 19.2 x - \frac{692}{3} = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $0.2$ out of $2.2 x^{2} - 19.2 x - \frac{692}{3}$ .

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Step 4.1.1

Factor $0.2$ out of $2.2 x^{2}$ .

$0.2 (11 x^{2}) - 19.2 x - \frac{692}{3} = 0$

Step 4.1.2

Factor $0.2$ out of $- 19.2 x$ .

$0.2 (11 x^{2}) + 0.2 (- 96 x) - \frac{692}{3} = 0$

Step 4.1.3

Factor $0.2$ out of $- \frac{692}{3}$ .

$0.2 (11 x^{2}) + 0.2 (- 96 x) + 0.2 (- 5 (\frac{692}{3})) = 0$

Step 4.1.4

Factor $0.2$ out of $0.2 (11 x^{2}) + 0.2 (- 96 x)$ .

$0.2 (11 x^{2} - 96 x) + 0.2 (- 5 (\frac{692}{3})) = 0$

Step 4.1.5

Factor $0.2$ out of $0.2 (11 x^{2} - 96 x) + 0.2 (- 5 (\frac{692}{3}))$ .

$0.2 (11 x^{2} - 96 x - 5 (\frac{692}{3})) = 0$

Step 4.2

Multiply $- 5 (\frac{692}{3})$ .

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Step 4.2.1

Combine $- 5$ and $\frac{692}{3}$ .

$0.2 (11 x^{2} - 96 x + \frac{- 5 \cdot 692}{3}) = 0$

Step 4.2.2

Multiply $- 5$ by $692$ .

$0.2 (11 x^{2} - 96 x + \frac{- 3460}{3}) = 0$

Step 4.3

Move the negative in front of the fraction.

$0.2 (11 x^{2} - 96 x - \frac{3460}{3}) = 0$

Step 5

Divide each term in $0.2 (11 x^{2} - 96 x - \frac{3460}{3}) = 0$ by $0.2$ and simplify.

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Step 5.1

Divide each term in $0.2 (11 x^{2} - 96 x - \frac{3460}{3}) = 0$ by $0.2$ .

$\frac{0.2 (11 x^{2} - 96 x - \frac{3460}{3})}{0.2} = \frac{0}{0.2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.2 (11 x^{2} - 96 x - \frac{3460}{3})}{0.2} = \frac{0}{0.2}$

Step 5.2.1.2

Divide $11 x^{2} - 96 x - \frac{3460}{3}$ by $1$ .

$11 x^{2} - 96 x - \frac{3460}{3} = \frac{0}{0.2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $0.2$ .

$11 x^{2} - 96 x - \frac{3460}{3} = 0$

Step 6

Multiply through by the least common denominator $3$ , then simplify.

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Step 6.1

Apply the distributive property.

$3 (11 x^{2}) + 3 (- 96 x) + 3 (- \frac{3460}{3}) = 0$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $11$ by $3$ .

$33 x^{2} + 3 (- 96 x) + 3 (- \frac{3460}{3}) = 0$

Step 6.2.2

Multiply $- 96$ by $3$ .

$33 x^{2} - 288 x + 3 (- \frac{3460}{3}) = 0$

Step 6.2.3

Cancel the common factor of $3$ .

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Step 6.2.3.1

Move the leading negative in $- \frac{3460}{3}$ into the numerator.

$33 x^{2} - 288 x + 3 (\frac{- 3460}{3}) = 0$

Step 6.2.3.2

Cancel the common factor.

$33 x^{2} - 288 x + 3 (\frac{- 3460}{3}) = 0$

Step 6.2.3.3

Rewrite the expression.

$33 x^{2} - 288 x - 3460 = 0$

Step 7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8

Substitute the values $a = 33$ , $b = - 288$ , and $c = - 3460$ into the quadratic formula and solve for $x$ .

$\frac{288 \pm \sqrt{{(- 288)}^{2} - 4 \cdot (33 \cdot - 3460)}}{2 \cdot 33}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 288$ to the power of $2$ .

$x = \frac{288 \pm \sqrt{82944 - 4 \cdot 33 \cdot - 3460}}{2 \cdot 33}$

Step 9.1.2

Multiply $- 4 \cdot 33 \cdot - 3460$ .

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Step 9.1.2.1

Multiply $- 4$ by $33$ .

$x = \frac{288 \pm \sqrt{82944 - 132 \cdot - 3460}}{2 \cdot 33}$

Step 9.1.2.2

Multiply $- 132$ by $- 3460$ .

$x = \frac{288 \pm \sqrt{82944 + 456720}}{2 \cdot 33}$

Step 9.1.3

Add $82944$ and $456720$ .

$x = \frac{288 \pm \sqrt{539664}}{2 \cdot 33}$

Step 9.1.4

Rewrite $539664$ as $4^{2} \cdot 33729$ .

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Step 9.1.4.1

Factor $16$ out of $539664$ .

$x = \frac{288 \pm \sqrt{16 (33729)}}{2 \cdot 33}$

Step 9.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{288 \pm \sqrt{4^{2} \cdot 33729}}{2 \cdot 33}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{288 \pm 4 \sqrt{33729}}{2 \cdot 33}$

Step 9.2

Multiply $2$ by $33$ .

$x = \frac{288 \pm 4 \sqrt{33729}}{66}$

Step 9.3

Simplify $\frac{288 \pm 4 \sqrt{33729}}{66}$ .

$x = \frac{144 \pm 2 \sqrt{33729}}{33}$

Step 10

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 288$ to the power of $2$ .

$x = \frac{288 \pm \sqrt{82944 - 4 \cdot 33 \cdot - 3460}}{2 \cdot 33}$

Step 10.1.2

Multiply $- 4 \cdot 33 \cdot - 3460$ .

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Step 10.1.2.1

Multiply $- 4$ by $33$ .

$x = \frac{288 \pm \sqrt{82944 - 132 \cdot - 3460}}{2 \cdot 33}$

Step 10.1.2.2

Multiply $- 132$ by $- 3460$ .

$x = \frac{288 \pm \sqrt{82944 + 456720}}{2 \cdot 33}$

Step 10.1.3

Add $82944$ and $456720$ .

$x = \frac{288 \pm \sqrt{539664}}{2 \cdot 33}$

Step 10.1.4

Rewrite $539664$ as $4^{2} \cdot 33729$ .

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Step 10.1.4.1

Factor $16$ out of $539664$ .

$x = \frac{288 \pm \sqrt{16 (33729)}}{2 \cdot 33}$

Step 10.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{288 \pm \sqrt{4^{2} \cdot 33729}}{2 \cdot 33}$

Step 10.1.5

Pull terms out from under the radical.

$x = \frac{288 \pm 4 \sqrt{33729}}{2 \cdot 33}$

Step 10.2

Multiply $2$ by $33$ .

$x = \frac{288 \pm 4 \sqrt{33729}}{66}$

Step 10.3

Simplify $\frac{288 \pm 4 \sqrt{33729}}{66}$ .

$x = \frac{144 \pm 2 \sqrt{33729}}{33}$

Step 10.4

Change the $\pm$ to $+$ .

$x = \frac{144 + 2 \sqrt{33729}}{33}$

Step 11

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 288$ to the power of $2$ .

$x = \frac{288 \pm \sqrt{82944 - 4 \cdot 33 \cdot - 3460}}{2 \cdot 33}$

Step 11.1.2

Multiply $- 4 \cdot 33 \cdot - 3460$ .

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Step 11.1.2.1

Multiply $- 4$ by $33$ .

$x = \frac{288 \pm \sqrt{82944 - 132 \cdot - 3460}}{2 \cdot 33}$

Step 11.1.2.2

Multiply $- 132$ by $- 3460$ .

$x = \frac{288 \pm \sqrt{82944 + 456720}}{2 \cdot 33}$

Step 11.1.3

Add $82944$ and $456720$ .

$x = \frac{288 \pm \sqrt{539664}}{2 \cdot 33}$

Step 11.1.4

Rewrite $539664$ as $4^{2} \cdot 33729$ .

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Step 11.1.4.1

Factor $16$ out of $539664$ .

$x = \frac{288 \pm \sqrt{16 (33729)}}{2 \cdot 33}$

Step 11.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{288 \pm \sqrt{4^{2} \cdot 33729}}{2 \cdot 33}$

Step 11.1.5

Pull terms out from under the radical.

$x = \frac{288 \pm 4 \sqrt{33729}}{2 \cdot 33}$

Step 11.2

Multiply $2$ by $33$ .

$x = \frac{288 \pm 4 \sqrt{33729}}{66}$

Step 11.3

Simplify $\frac{288 \pm 4 \sqrt{33729}}{66}$ .

$x = \frac{144 \pm 2 \sqrt{33729}}{33}$

Step 11.4

Change the $\pm$ to $-$ .

$x = \frac{144 - 2 \sqrt{33729}}{33}$

Step 12

The final answer is the combination of both solutions.

$x = \frac{144 + 2 \sqrt{33729}}{33}, \frac{144 - 2 \sqrt{33729}}{33}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{144 + 2 \sqrt{33729}}{33}, \frac{144 - 2 \sqrt{33729}}{33}$

Decimal Form:

$x = 15.49421618 \dots, - 6.76694345 \dots$