Algebra Examples

$4.1 \cdot 10^{- 4} = \frac{{(2 x)}^{2}}{(0.25 - x) (0.43 - x)}$

Step 1

Rewrite the equation as $\frac{{(2 x)}^{2}}{(0.25 - x) (0.43 - x)} = 4.1 \cdot 10^{- 4}$ .

$\frac{{(2 x)}^{2}}{(0.25 - x) (0.43 - x)} = 4.1 \cdot 10^{- 4}$

Step 2

Factor each term.

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Step 2.1

Apply the product rule to $2 x$ .

$\frac{2^{2} x^{2}}{(0.25 - x) (0.43 - x)} = 4.1 \cdot 10^{- 4}$

Step 2.2

Factor $0.25$ out of $0.25 - x$ .

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Step 2.2.1

Factor $0.25$ out of $0.25$ .

$\frac{2^{2} x^{2}}{(0.25 (1) - x) (0.43 - x)} = 4.1 \cdot 10^{- 4}$

Step 2.2.2

Factor $0.25$ out of $- x$ .

$\frac{2^{2} x^{2}}{(0.25 (1) + 0.25 (- 4 x)) (0.43 - x)} = 4.1 \cdot 10^{- 4}$

Step 2.2.3

Factor $0.25$ out of $0.25 (1) + 0.25 (- 4 x)$ .

$\frac{2^{2} x^{2}}{0.25 (1 - 4 x) (0.43 - x)} = 4.1 \cdot 10^{- 4}$

Step 2.3

Factor.

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Step 2.3.1

Factor $0.01$ out of $0.43 - x$ .

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Step 2.3.1.1

Factor $0.01$ out of $0.43$ .

$\frac{2^{2} x^{2}}{0.25 (1 - 4 x) (0.01 (43) - x)} = 4.1 \cdot 10^{- 4}$

Step 2.3.1.2

Factor $0.01$ out of $- x$ .

$\frac{2^{2} x^{2}}{0.25 (1 - 4 x) (0.01 (43) + 0.01 (- 100 x))} = 4.1 \cdot 10^{- 4}$

Step 2.3.1.3

Factor $0.01$ out of $0.01 (43) + 0.01 (- 100 x)$ .

$\frac{2^{2} x^{2}}{0.25 (1 - 4 x) (0.01 (43 - 100 x))} = 4.1 \cdot 10^{- 4}$

Step 2.3.2

Remove unnecessary parentheses.

$\frac{2^{2} x^{2}}{0.25 (1 - 4 x) \cdot 0.01 (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 2.4

Multiply $0.01$ by $0.25$ .

$\frac{2^{2} x^{2}}{0.0025 (1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 2.5

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

$\frac{4 x^{2}}{0.0025 (1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 2.6

Factor $4$ out of $4 x^{2}$ .

$\frac{4 (x^{2})}{0.0025 (1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 2.7

Factor $0.0025$ out of $0.0025 (1 - 4 x) (43 - 100 x)$ .

$\frac{4 (x^{2})}{0.0025 ((1 - 4 x) (43 - 100 x))} = 4.1 \cdot 10^{- 4}$

Step 2.8

Separate fractions.

$\frac{4}{0.0025} \cdot \frac{x^{2}}{(1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 2.9

Divide $4$ by $0.0025$ .

$1600 \frac{x^{2}}{(1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 2.10

Combine $1600$ and $\frac{x^{2}}{(1 - 4 x) (43 - 100 x)}$ .

$\frac{1600 x^{2}}{(1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(1 - 4 x) (43 - 100 x), 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$(1 - 4 x) (43 - 100 x)$

Step 4

Multiply each term in $\frac{1600 x^{2}}{(1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$ by $(1 - 4 x) (43 - 100 x)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{1600 x^{2}}{(1 - 4 x) (43 - 100 x)} = 4.1 \cdot 10^{- 4}$ by $(1 - 4 x) (43 - 100 x)$ .

$\frac{1600 x^{2}}{(1 - 4 x) (43 - 100 x)} ((1 - 4 x) (43 - 100 x)) = 4.1 \cdot 10^{- 4} ((1 - 4 x) (43 - 100 x))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $(1 - 4 x) (43 - 100 x)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{1600 x^{2}}{(1 - 4 x) (43 - 100 x)} ((1 - 4 x) (43 - 100 x)) = 4.1 \cdot 10^{- 4} ((1 - 4 x) (43 - 100 x))$

Step 4.2.1.2

Rewrite the expression.

$1600 x^{2} = 4.1 \cdot 10^{- 4} ((1 - 4 x) (43 - 100 x))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Expand $(1 - 4 x) (43 - 100 x)$ using the FOIL Method.

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Step 4.3.1.1

Apply the distributive property.

$1600 x^{2} = 4.1 \cdot 10^{- 4} (1 (43 - 100 x) - 4 x (43 - 100 x))$

Step 4.3.1.2

Apply the distributive property.

$1600 x^{2} = 4.1 \cdot 10^{- 4} (1 \cdot 43 + 1 (- 100 x) - 4 x (43 - 100 x))$

Step 4.3.1.3

Apply the distributive property.

$1600 x^{2} = 4.1 \cdot 10^{- 4} (1 \cdot 43 + 1 (- 100 x) - 4 x \cdot 43 - 4 x (- 100 x))$

Step 4.3.2

Simplify and combine like terms.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $43$ by $1$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 + 1 (- 100 x) - 4 x \cdot 43 - 4 x (- 100 x))$

Step 4.3.2.1.2

Multiply $- 100 x$ by $1$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 100 x - 4 x \cdot 43 - 4 x (- 100 x))$

Step 4.3.2.1.3

Multiply $43$ by $- 4$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 100 x - 172 x - 4 x (- 100 x))$

Step 4.3.2.1.4

Rewrite using the commutative property of multiplication.

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 100 x - 172 x - 4 \cdot - 100 x \cdot x)$

Step 4.3.2.1.5

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

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Step 4.3.2.1.5.1

Move $x$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 100 x - 172 x - 4 \cdot - 100 (x \cdot x))$

Step 4.3.2.1.5.2

Multiply $x$ by $x$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 100 x - 172 x - 4 \cdot - 100 x^{2})$

Step 4.3.2.1.6

Multiply $- 4$ by $- 100$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 100 x - 172 x + 400 x^{2})$

Step 4.3.2.2

Subtract $172 x$ from $- 100 x$ .

$1600 x^{2} = 4.1 \cdot 10^{- 4} (43 - 272 x + 400 x^{2})$

Step 4.3.3

Apply the distributive property.

$1600 x^{2} = 4.1 \cdot 10^{- 4} \cdot 43 + 4.1 \cdot 10^{- 4} (- 272 x) + 4.1 \cdot 10^{- 4} (400 x^{2})$

Step 4.3.4

Simplify.

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Step 4.3.4.1

Multiply $4.1$ by $43$ .

$1600 x^{2} = 176.3 \cdot 10^{- 4} + 4.1 \cdot 10^{- 4} (- 272 x) + 4.1 \cdot 10^{- 4} (400 x^{2})$

Step 4.3.4.2

Multiply $- 272$ by $4.1$ .

$1600 x^{2} = 176.3 \cdot 10^{- 4} - 1115.2 \cdot 10^{- 4} x + 4.1 \cdot 10^{- 4} (400 x^{2})$

Step 4.3.4.3

Multiply $400$ by $4.1$ .

$1600 x^{2} = 176.3 \cdot 10^{- 4} - 1115.2 \cdot 10^{- 4} x + 1640 \cdot 10^{- 4} x^{2}$

Step 4.3.5

Simplify each term.

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Step 4.3.5.1

Move the decimal point in $176.3$ to the left by $2$ places and increase the power of $10^{- 4}$ by $2$ .

$1600 x^{2} = 1.763 \cdot 10^{- 2} - 1115.2 \cdot 10^{- 4} x + 1640 \cdot 10^{- 4} x^{2}$

Step 4.3.5.2

Move the decimal point in $- 1115.2$ to the left by $3$ places and increase the power of $10^{- 4}$ by $3$ .

$1600 x^{2} = 1.763 \cdot 10^{- 2} - 1.1152 \cdot 10^{- 1} x + 1640 \cdot 10^{- 4} x^{2}$

Step 4.3.5.3

Move the decimal point in $1640$ to the left by $3$ places and increase the power of $10^{- 4}$ by $3$ .

$1600 x^{2} = 1.763 \cdot 10^{- 2} - 1.1152 \cdot 10^{- 1} x + 1.64 \cdot 10^{- 1} x^{2}$

Step 4.3.6

Reorder factors in $1.763 \cdot 10^{- 2} - 1.1152 \cdot 10^{- 1} x + 1.64 \cdot 10^{- 1} x^{2}$ .

$1600 x^{2} = 1.763 \cdot 10^{- 2} - 1.1152 x \cdot 10^{- 1} + 1.64 x^{2} \cdot 10^{- 1}$

Step 5

Solve the equation.

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Step 5.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$1.763 \cdot 10^{- 2} - 1.1152 x \cdot 10^{- 1} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2

Simplify each term.

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Step 5.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1.763 \cdot 10^{- 2} - 1.1152 x \cdot \frac{1}{10} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.2

Multiply $- 1.1152 x \frac{1}{10}$ .

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Step 5.2.2.1

Combine $- 1.1152$ and $\frac{1}{10}$ .

$1.763 \cdot 10^{- 2} + x \frac{- 1.1152}{10} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.2.2

Combine $x$ and $\frac{- 1.1152}{10}$ .

$1.763 \cdot 10^{- 2} + \frac{x \cdot - 1.1152}{10} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.3

Move $- 1.1152$ to the left of $x$ .

$1.763 \cdot 10^{- 2} + \frac{- 1.1152 \cdot x}{10} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.4

Move the negative in front of the fraction.

$1.763 \cdot 10^{- 2} - \frac{1.1152 \cdot x}{10} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.5

Factor $1.1152$ out of $1.1152 \cdot x$ .

$1.763 \cdot 10^{- 2} - \frac{1.1152 (x)}{10} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.6

Factor $10$ out of $10$ .

$1.763 \cdot 10^{- 2} - \frac{1.1152 (x)}{10 (1)} + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.7

Separate fractions.

$1.763 \cdot 10^{- 2} - (\frac{1.1152}{10} \cdot \frac{x}{1}) + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.8

Divide $1.1152$ by $10$ .

$1.763 \cdot 10^{- 2} - (0.11152 \frac{x}{1}) + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.9

Divide $x$ by $1$ .

$1.763 \cdot 10^{- 2} - (0.11152 x) + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.10

Multiply $0.11152$ by $- 1$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + 1.64 x^{2} \cdot 10^{- 1} = 1600 x^{2}$

Step 5.2.11

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + 1.64 x^{2} \cdot \frac{1}{10} = 1600 x^{2}$

Step 5.2.12

Multiply $1.64 x^{2} \frac{1}{10}$ .

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Step 5.2.12.1

Combine $1.64$ and $\frac{1}{10}$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + x^{2} \frac{1.64}{10} = 1600 x^{2}$

Step 5.2.12.2

Combine $x^{2}$ and $\frac{1.64}{10}$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + \frac{x^{2} \cdot 1.64}{10} = 1600 x^{2}$

Step 5.2.13

Move $1.64$ to the left of $x^{2}$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + \frac{1.64 \cdot x^{2}}{10} = 1600 x^{2}$

Step 5.2.14

Factor $1.64$ out of $1.64 \cdot x^{2}$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + \frac{1.64 (x^{2})}{10} = 1600 x^{2}$

Step 5.2.15

Factor $10$ out of $10$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + \frac{1.64 (x^{2})}{10 (1)} = 1600 x^{2}$

Step 5.2.16

Separate fractions.

$1.763 \cdot 10^{- 2} - 0.11152 x + \frac{1.64}{10} \cdot \frac{x^{2}}{1} = 1600 x^{2}$

Step 5.2.17

Divide $1.64$ by $10$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + 0.164 \frac{x^{2}}{1} = 1600 x^{2}$

Step 5.2.18

Divide $x^{2}$ by $1$ .

$1.763 \cdot 10^{- 2} - 0.11152 x + 0.164 x^{2} = 1600 x^{2}$

Step 5.3

Move all terms containing $x$ to the left side of the equation.

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Step 5.3.1

Subtract $1600 x^{2}$ from both sides of the equation.

$1.763 \cdot 10^{- 2} - 0.11152 x + 0.164 x^{2} - 1600 x^{2} = 0$

Step 5.3.2

Subtract $1600 x^{2}$ from $0.164 x^{2}$ .

$1.763 \cdot 10^{- 2} - 0.11152 x - 1599.836 x^{2} = 0$

Step 5.4

Factor the left side of the equation.

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Step 5.4.1

Factor $- 0.00004$ out of $1.763 \cdot 10^{- 2} - 0.11152 x - 1599.836 x^{2}$ .

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Step 5.4.1.1

Reorder the expression.

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Step 5.4.1.1.1

Move $1.763 \cdot 10^{- 2}$ .

$- 0.11152 x - 1599.836 x^{2} + 1.763 \cdot 10^{- 2} = 0$

Step 5.4.1.1.2

Reorder $- 0.11152 x$ and $- 1599.836 x^{2}$ .

$- 1599.836 x^{2} - 0.11152 x + 1.763 \cdot 10^{- 2} = 0$

Step 5.4.1.2

Factor $- 0.00004$ out of $- 1599.836 x^{2}$ .

$- 0.00004 (39995900 x^{2}) - 0.11152 x + 1.763 \cdot 10^{- 2} = 0$

Step 5.4.1.3

Factor $- 0.00004$ out of $- 0.11152 x$ .

$- 0.00004 (39995900 x^{2}) - 0.00004 (2788 x) + 1.763 \cdot 10^{- 2} = 0$

Step 5.4.1.4

Factor $- 0.00004$ out of $1.763 \cdot 10^{- 2}$ .

$- 0.00004 (39995900 x^{2}) - 0.00004 (2788 x) - 0.00004 \cdot - 44075 \cdot 10^{- 2} = 0$

Step 5.4.1.5

Factor $- 0.00004$ out of $- 0.00004 (39995900 x^{2}) - 0.00004 (2788 x)$ .

$- 0.00004 (39995900 x^{2} + 2788 x) - 0.00004 \cdot - 44075 \cdot 10^{- 2} = 0$

Step 5.4.1.6

Factor $- 0.00004$ out of $- 0.00004 (39995900 x^{2} + 2788 x) - 0.00004 \cdot - 44075 \cdot 10^{- 2}$ .

$- 0.00004 (39995900 x^{2} + 2788 x - 44075 \cdot 10^{- 2}) = 0$

Step 5.4.2

Move the decimal point in $- 44075$ to the left by $4$ places and increase the power of $10^{- 2}$ by $4$ .

$- 0.00004 (39995900 x^{2} + 2788 x - 4.4075 \cdot 10^{2}) = 0$

Step 5.4.3

Reorder terms.

$- 0.00004 (39995900 x^{2} - 4.4075 \cdot 10^{2} + 2788 x) = 0$

Step 5.4.4

Factor.

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Step 5.4.4.1

Rewrite $39995900 x^{2} - 4.4075 \cdot 10^{2} + 2788 x$ in a factored form.

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Step 5.4.4.1.1

Factor $0.0025$ out of $39995900 x^{2} - 4.4075 \cdot 10^{2} + 2788 x$ .

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Step 5.4.4.1.1.1

Factor $0.0025$ out of $39995900 x^{2}$ .

$- 0.00004 (0.0025 (15998360000 x^{2}) - 4.4075 \cdot 10^{2} + 2788 x) = 0$

Step 5.4.4.1.1.2

Factor $0.0025$ out of $- 4.4075 \cdot 10^{2}$ .

$- 0.00004 (0.0025 (15998360000 x^{2}) + 0.0025 \cdot - 1763 \cdot 10^{2} + 2788 x) = 0$

Step 5.4.4.1.1.3

Factor $0.0025$ out of $2788 x$ .

$- 0.00004 (0.0025 (15998360000 x^{2}) + 0.0025 \cdot - 1763 \cdot 10^{2} + 0.0025 (1115200 x)) = 0$

Step 5.4.4.1.1.4

Factor $0.0025$ out of $0.0025 (15998360000 x^{2}) + 0.0025 \cdot - 1763 \cdot 10^{2}$ .

$- 0.00004 (0.0025 (15998360000 x^{2} - 1763 \cdot 10^{2}) + 0.0025 (1115200 x)) = 0$

Step 5.4.4.1.1.5

Factor $0.0025$ out of $0.0025 (15998360000 x^{2} - 1763 \cdot 10^{2}) + 0.0025 (1115200 x)$ .

$- 0.00004 (0.0025 (15998360000 x^{2} - 1763 \cdot 10^{2} + 1115200 x)) = 0$

Step 5.4.4.1.2

Move the decimal point in $- 1763$ to the left by $3$ places and increase the power of $10^{2}$ by $3$ .

$- 0.00004 (0.0025 (15998360000 x^{2} - 1.763 \cdot 10^{5} + 1115200 x)) = 0$

Step 5.4.4.1.3

Reorder terms.

$- 0.00004 (0.0025 (- 1.763 \cdot 10^{5} + 15998360000 x^{2} + 1115200 x)) = 0$

Step 5.4.4.1.4

Factor.

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Step 5.4.4.1.4.1

Rewrite $- 1.763 \cdot 10^{5} + 15998360000 x^{2} + 1115200 x$ in a factored form.

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Step 5.4.4.1.4.1.1

Factor $0.001$ out of $- 1.763 \cdot 10^{5} + 15998360000 x^{2} + 1115200 x$ .

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Step 5.4.4.1.4.1.1.1

Factor $0.001$ out of $- 1.763 \cdot 10^{5}$ .

$- 0.00004 (0.0025 (0.001 \cdot - 1763 \cdot 10^{5} + 15998360000 x^{2} + 1115200 x)) = 0$

Step 5.4.4.1.4.1.1.2

Factor $0.001$ out of $15998360000 x^{2}$ .

$- 0.00004 (0.0025 (0.001 \cdot - 1763 \cdot 10^{5} + 0.001 (15998360000000 x^{2}) + 1115200 x)) = 0$

Step 5.4.4.1.4.1.1.3

Factor $0.001$ out of $1115200 x$ .

$- 0.00004 (0.0025 (0.001 \cdot - 1763 \cdot 10^{5} + 0.001 (15998360000000 x^{2}) + 0.001 (1115200000 x))) = 0$

Step 5.4.4.1.4.1.1.4

Factor $0.001$ out of $0.001 \cdot - 1763 \cdot 10^{5} + 0.001 (15998360000000 x^{2})$ .

$- 0.00004 (0.0025 (0.001 \cdot (- 1763 \cdot 10^{5} + 15998360000000 x^{2}) + 0.001 (1115200000 x))) = 0$

Step 5.4.4.1.4.1.1.5

Factor $0.001$ out of $0.001 \cdot (- 1763 \cdot 10^{5} + 15998360000000 x^{2}) + 0.001 (1115200000 x)$ .

$- 0.00004 (0.0025 (0.001 (- 1763 \cdot 10^{5} + 15998360000000 x^{2} + 1115200000 x))) = 0$

Step 5.4.4.1.4.1.2

Let $u = - 1763$ . Substitute $u$ for all occurrences of $- 1763$ .

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Step 5.4.4.1.4.1.2.1

Raise $10$ to the power of $5$ .

$- 0.00004 (0.0025 (0.001 (u \cdot 100000 + 15998360000000 x^{2} + 1115200000 x))) = 0$

Step 5.4.4.1.4.1.2.2

Move $100000$ to the left of $u$ .

$- 0.00004 (0.0025 (0.001 (100000 u + 15998360000000 x^{2} + 1115200000 x))) = 0$

Step 5.4.4.1.4.1.3

Factor $100000$ out of $100000 u + 15998360000000 x^{2} + 1115200000 x$ .

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Step 5.4.4.1.4.1.3.1

Factor $100000$ out of $100000 u$ .

$- 0.00004 (0.0025 (0.001 (100000 (u) + 15998360000000 x^{2} + 1115200000 x))) = 0$

Step 5.4.4.1.4.1.3.2

Factor $100000$ out of $15998360000000 x^{2}$ .

$- 0.00004 (0.0025 (0.001 (100000 (u) + 100000 (159983600 x^{2}) + 1115200000 x))) = 0$

Step 5.4.4.1.4.1.3.3

Factor $100000$ out of $1115200000 x$ .

$- 0.00004 (0.0025 (0.001 (100000 (u) + 100000 (159983600 x^{2}) + 100000 (11152 x)))) = 0$

Step 5.4.4.1.4.1.3.4

Factor $100000$ out of $100000 (u) + 100000 (159983600 x^{2})$ .

$- 0.00004 (0.0025 (0.001 (100000 (u + 159983600 x^{2}) + 100000 (11152 x)))) = 0$

Step 5.4.4.1.4.1.3.5

Factor $100000$ out of $100000 (u + 159983600 x^{2}) + 100000 (11152 x)$ .

$- 0.00004 (0.0025 (0.001 (100000 (u + 159983600 x^{2} + 11152 x)))) = 0$

Step 5.4.4.1.4.1.4

Replace all occurrences of $u$ with $- 1763$ .

$- 0.00004 (0.0025 (0.001 (100000 (- 1763 + 159983600 x^{2} + 11152 x)))) = 0$

Step 5.4.4.1.4.1.5

Factor.

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Step 5.4.4.1.4.1.5.1

Reorder terms.

$- 0.00004 (0.0025 (0.001 (100000 (159983600 x^{2} + 11152 x - 1763)))) = 0$

Step 5.4.4.1.4.1.5.2

Remove unnecessary parentheses.

$- 0.00004 (0.0025 (0.001 \cdot (100000 (159983600 x^{2} + 11152 x - 1763)))) = 0$

Step 5.4.4.1.4.1.6

Multiply $0.001$ by $100000$ .

$- 0.00004 (0.0025 (100 (159983600 x^{2} + 11152 x - 1763))) = 0$

Step 5.4.4.1.4.2

Remove unnecessary parentheses.

$- 0.00004 (0.0025 \cdot (100 (159983600 x^{2} + 11152 x - 1763))) = 0$

Step 5.4.4.1.5

Multiply $0.0025$ by $100$ .

$- 0.00004 (0.25 (159983600 x^{2} + 11152 x - 1763)) = 0$

Step 5.4.4.2

Remove unnecessary parentheses.

$- 0.00004 \cdot (0.25 (159983600 x^{2} + 11152 x - 1763)) = 0$

Step 5.4.5

Multiply $- 0.00004$ by $0.25$ .

$- 0.00001 (159983600 x^{2} + 11152 x - 1763) = 0$

Step 5.5

Divide each term in $- 0.00001 (159983600 x^{2} + 11152 x - 1763) = 0$ by $- 0.00001$ and simplify.

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Step 5.5.1

Divide each term in $- 0.00001 (159983600 x^{2} + 11152 x - 1763) = 0$ by $- 0.00001$ .

$\frac{- 0.00001 (159983600 x^{2} + 11152 x - 1763)}{- 0.00001} = \frac{0}{- 0.00001}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 0.00001$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{- 0.00001 (159983600 x^{2} + 11152 x - 1763)}{- 0.00001} = \frac{0}{- 0.00001}$

Step 5.5.2.1.2

Divide $159983600 x^{2} + 11152 x - 1763$ by $1$ .

$159983600 x^{2} + 11152 x - 1763 = \frac{0}{- 0.00001}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $0$ by $- 0.00001$ .

$159983600 x^{2} + 11152 x - 1763 = 0$

Step 5.6

Use the quadratic formula to find the solutions.

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

Step 5.7

Substitute the values $a = 159983600$ , $b = 11152$ , and $c = - 1763$ into the quadratic formula and solve for $x$ .

$\frac{- 11152 \pm \sqrt{11152^{2} - 4 \cdot (159983600 \cdot - 1763)}}{2 \cdot 159983600}$

Step 5.8

Simplify.

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Step 5.8.1

Simplify the numerator.

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Step 5.8.1.1

Raise $11152$ to the power of $2$ .

$x = \frac{- 11152 \pm \sqrt{124367104 - 4 \cdot 159983600 \cdot - 1763}}{2 \cdot 159983600}$

Step 5.8.1.2

Multiply $- 4 \cdot 159983600 \cdot - 1763$ .

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Step 5.8.1.2.1

Multiply $- 4$ by $159983600$ .

$x = \frac{- 11152 \pm \sqrt{124367104 - 639934400 \cdot - 1763}}{2 \cdot 159983600}$

Step 5.8.1.2.2

Multiply $- 639934400$ by $- 1763$ .

$x = \frac{- 11152 \pm \sqrt{124367104 + 1128204347200}}{2 \cdot 159983600}$

Step 5.8.1.3

Add $124367104$ and $1128204347200$ .

$x = \frac{- 11152 \pm \sqrt{1128328714304}}{2 \cdot 159983600}$

Step 5.8.1.4

Rewrite $1128328714304$ as $8^{2} \cdot 17630136161$ .

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Step 5.8.1.4.1

Factor $64$ out of $1128328714304$ .

$x = \frac{- 11152 \pm \sqrt{64 (17630136161)}}{2 \cdot 159983600}$

Step 5.8.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 11152 \pm \sqrt{8^{2} \cdot 17630136161}}{2 \cdot 159983600}$

Step 5.8.1.5

Pull terms out from under the radical.

$x = \frac{- 11152 \pm 8 \sqrt{17630136161}}{2 \cdot 159983600}$

Step 5.8.2

Multiply $2$ by $159983600$ .

$x = \frac{- 11152 \pm 8 \sqrt{17630136161}}{319967200}$

Step 5.8.3

Simplify $\frac{- 11152 \pm 8 \sqrt{17630136161}}{319967200}$ .

$x = \frac{- 1394 \pm \sqrt{17630136161}}{39995900}$

Step 5.9

The final answer is the combination of both solutions.

$x = - \frac{1394 - \sqrt{17630136161}}{39995900}, - \frac{1394 + \sqrt{17630136161}}{39995900}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1394 - \sqrt{17630136161}}{39995900}, - \frac{1394 + \sqrt{17630136161}}{39995900}$

Decimal Form:

$x = 0.00328494 \dots, - 0.00335465 \dots$