Finite Math Examples

${(1 + x)}^{2} (1 + 0.5 x) = 1.2705$

Step 1

Simplify ${(1 + x)}^{2} (1 + 0.5 x)$ .

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Step 1.1

Apply the distributive property.

${(1 + x)}^{2} \cdot 1 + {(1 + x)}^{2} (0.5 x) = 1.2705$

Step 1.2

Simplify the expression.

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Step 1.2.1

Multiply ${(1 + x)}^{2}$ by $1$ .

${(1 + x)}^{2} + {(1 + x)}^{2} (0.5 x) = 1.2705$

Step 1.2.2

Rewrite using the commutative property of multiplication.

${(1 + x)}^{2} + 0.5 {(1 + x)}^{2} x = 1.2705$

Step 1.2.3

Reorder factors in ${(1 + x)}^{2} + 0.5 {(1 + x)}^{2} x$ .

${(1 + x)}^{2} + 0.5 x {(1 + x)}^{2} = 1.2705$

Step 2

Subtract $1.2705$ from both sides of the equation.

${(1 + x)}^{2} + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3

Simplify ${(1 + x)}^{2} + 0.5 x {(1 + x)}^{2} - 1.2705$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$(1 + x) (1 + x) + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.2

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

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Step 3.1.2.1

Apply the distributive property.

$1 (1 + x) + x (1 + x) + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.2.2

Apply the distributive property.

$1 \cdot 1 + 1 x + x (1 + x) + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.2.3

Apply the distributive property.

$1 \cdot 1 + 1 x + x \cdot 1 + x \cdot x + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

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

Multiply $1$ by $1$ .

$1 + 1 x + x \cdot 1 + x \cdot x + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.3.1.2

Multiply $x$ by $1$ .

$1 + x + x \cdot 1 + x \cdot x + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.3.1.3

Multiply $x$ by $1$ .

$1 + x + x + x \cdot x + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.3.1.4

Multiply $x$ by $x$ .

$1 + x + x + x^{2} + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.3.2

Add $x$ and $x$ .

$1 + 2 x + x^{2} + 0.5 x {(1 + x)}^{2} - 1.2705 = 0$

Step 3.1.4

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

$1 + 2 x + x^{2} + 0.5 x ((1 + x) (1 + x)) - 1.2705 = 0$

Step 3.1.5

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

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Step 3.1.5.1

Apply the distributive property.

$1 + 2 x + x^{2} + 0.5 x (1 (1 + x) + x (1 + x)) - 1.2705 = 0$

Step 3.1.5.2

Apply the distributive property.

$1 + 2 x + x^{2} + 0.5 x (1 \cdot 1 + 1 x + x (1 + x)) - 1.2705 = 0$

Step 3.1.5.3

Apply the distributive property.

$1 + 2 x + x^{2} + 0.5 x (1 \cdot 1 + 1 x + x \cdot 1 + x \cdot x) - 1.2705 = 0$

Step 3.1.6

Simplify and combine like terms.

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Step 3.1.6.1

Simplify each term.

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Step 3.1.6.1.1

Multiply $1$ by $1$ .

$1 + 2 x + x^{2} + 0.5 x (1 + 1 x + x \cdot 1 + x \cdot x) - 1.2705 = 0$

Step 3.1.6.1.2

Multiply $x$ by $1$ .

$1 + 2 x + x^{2} + 0.5 x (1 + x + x \cdot 1 + x \cdot x) - 1.2705 = 0$

Step 3.1.6.1.3

Multiply $x$ by $1$ .

$1 + 2 x + x^{2} + 0.5 x (1 + x + x + x \cdot x) - 1.2705 = 0$

Step 3.1.6.1.4

Multiply $x$ by $x$ .

$1 + 2 x + x^{2} + 0.5 x (1 + x + x + x^{2}) - 1.2705 = 0$

Step 3.1.6.2

Add $x$ and $x$ .

$1 + 2 x + x^{2} + 0.5 x (1 + 2 x + x^{2}) - 1.2705 = 0$

Step 3.1.7

Apply the distributive property.

$1 + 2 x + x^{2} + 0.5 x \cdot 1 + 0.5 x (2 x) + 0.5 x \cdot x^{2} - 1.2705 = 0$

Step 3.1.8

Simplify.

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Step 3.1.8.1

Multiply $0.5$ by $1$ .

$1 + 2 x + x^{2} + 0.5 x + 0.5 x (2 x) + 0.5 x \cdot x^{2} - 1.2705 = 0$

Step 3.1.8.2

Rewrite using the commutative property of multiplication.

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 x \cdot x + 0.5 x \cdot x^{2} - 1.2705 = 0$

Step 3.1.8.3

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

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Step 3.1.8.3.1

Move $x^{2}$ .

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 x \cdot x + 0.5 (x^{2} x) - 1.2705 = 0$

Step 3.1.8.3.2

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

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Step 3.1.8.3.2.1

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

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 x \cdot x + 0.5 (x^{2} x^{1}) - 1.2705 = 0$

Step 3.1.8.3.2.2

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

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 x \cdot x + 0.5 x^{2 + 1} - 1.2705 = 0$

Step 3.1.8.3.3

Add $2$ and $1$ .

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 x \cdot x + 0.5 x^{3} - 1.2705 = 0$

Step 3.1.9

Simplify each term.

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Step 3.1.9.1

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

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Step 3.1.9.1.1

Move $x$ .

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 (x \cdot x) + 0.5 x^{3} - 1.2705 = 0$

Step 3.1.9.1.2

Multiply $x$ by $x$ .

$1 + 2 x + x^{2} + 0.5 x + 0.5 \cdot 2 x^{2} + 0.5 x^{3} - 1.2705 = 0$

Step 3.1.9.2

Multiply $0.5$ by $2$ .

$1 + 2 x + x^{2} + 0.5 x + 1 x^{2} + 0.5 x^{3} - 1.2705 = 0$

Step 3.1.9.3

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

$1 + 2 x + x^{2} + 0.5 x + x^{2} + 0.5 x^{3} - 1.2705 = 0$

Step 3.2

Subtract $1.2705$ from $1$ .

$2 x + x^{2} + 0.5 x + x^{2} + 0.5 x^{3} - 0.2705 = 0$

Step 3.3

Add $2 x$ and $0.5 x$ .

$x^{2} + 2.5 x + x^{2} + 0.5 x^{3} - 0.2705 = 0$

Step 3.4

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 2.5 x + 0.5 x^{3} - 0.2705 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $0.0005$ out of $2 x^{2} + 2.5 x + 0.5 x^{3} - 0.2705$ .

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Step 4.1.1

Factor $0.0005$ out of $2 x^{2}$ .

$0.0005 (4000 x^{2}) + 2.5 x + 0.5 x^{3} - 0.2705 = 0$

Step 4.1.2

Factor $0.0005$ out of $2.5 x$ .

$0.0005 (4000 x^{2}) + 0.0005 (5000 x) + 0.5 x^{3} - 0.2705 = 0$

Step 4.1.3

Factor $0.0005$ out of $0.5 x^{3}$ .

$0.0005 (4000 x^{2}) + 0.0005 (5000 x) + 0.0005 (1000 x^{3}) - 0.2705 = 0$

Step 4.1.4

Factor $0.0005$ out of $- 0.2705$ .

$0.0005 (4000 x^{2}) + 0.0005 (5000 x) + 0.0005 (1000 x^{3}) + 0.0005 (- 541) = 0$

Step 4.1.5

Factor $0.0005$ out of $0.0005 (4000 x^{2}) + 0.0005 (5000 x)$ .

$0.0005 (4000 x^{2} + 5000 x) + 0.0005 (1000 x^{3}) + 0.0005 (- 541) = 0$

Step 4.1.6

Factor $0.0005$ out of $0.0005 (4000 x^{2} + 5000 x) + 0.0005 (1000 x^{3})$ .

$0.0005 (4000 x^{2} + 5000 x + 1000 x^{3}) + 0.0005 (- 541) = 0$

Step 4.1.7

Factor $0.0005$ out of $0.0005 (4000 x^{2} + 5000 x + 1000 x^{3}) + 0.0005 (- 541)$ .

$0.0005 (4000 x^{2} + 5000 x + 1000 x^{3} - 541) = 0$

Step 4.2

Reorder terms.

$0.0005 (1000 x^{3} + 4000 x^{2} + 5000 x - 541) = 0$

Step 4.3

Factor.

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Step 4.3.1

Factor $1000 x^{3} + 4000 x^{2} + 5000 x - 541$ using the rational roots test.

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Step 4.3.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 541$

$q = \pm 1, \pm 1000, \pm 2, \pm 500, \pm 4, \pm 250, \pm 5, \pm 200, \pm 8, \pm 125, \pm 10, \pm 100, \pm 20, \pm 50, \pm 25, \pm 40$

Step 4.3.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.001, \pm 0.5, \pm 0.002, \pm 0.25, \pm 0.004, \pm 0.2, \pm 0.005, \pm 0.125, \pm 0.008, \pm 0.1, \pm 0.01, \pm 0.05, \pm 0.02, \pm 0.04, \pm 0.025, \pm 541, \pm 0.541, \pm 270.5, \pm 1.082, \pm 135.25, \pm 2.164, \pm 108.2, \pm 2.705, \pm 67.625, \pm 4.328, \pm 54.1, \pm 5.41, \pm 27.05, \pm 10.82, \pm 21.64, \pm 13.525$

Step 4.3.1.3

Substitute $0.1$ and simplify the expression. In this case, the expression is equal to $0$ so $0.1$ is a root of the polynomial.

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Step 4.3.1.3.1

Substitute $0.1$ into the polynomial.

$1000 \cdot {0.1}^{3} + 4000 \cdot {0.1}^{2} + 5000 \cdot 0.1 - 541$

Step 4.3.1.3.2

Raise $0.1$ to the power of $3$ .

$1000 \cdot 0.001 + 4000 \cdot {0.1}^{2} + 5000 \cdot 0.1 - 541$

Step 4.3.1.3.3

Multiply $1000$ by $0.001$ .

$1 + 4000 \cdot {0.1}^{2} + 5000 \cdot 0.1 - 541$

Step 4.3.1.3.4

Raise $0.1$ to the power of $2$ .

$1 + 4000 \cdot 0.01 + 5000 \cdot 0.1 - 541$

Step 4.3.1.3.5

Multiply $4000$ by $0.01$ .

$1 + 40 + 5000 \cdot 0.1 - 541$

Step 4.3.1.3.6

Add $1$ and $40$ .

$41 + 5000 \cdot 0.1 - 541$

Step 4.3.1.3.7

Multiply $5000$ by $0.1$ .

$41 + 500 - 541$

Step 4.3.1.3.8

Add $41$ and $500$ .

$541 - 541$

Step 4.3.1.3.9

Subtract $541$ from $541$ .

$0$

Step 4.3.1.4

Since $0.1$ is a known root, divide the polynomial by $10 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{1000 x^{3} + 4000 x^{2} + 5000 x - 541}{10 x - 1}$

Step 4.3.1.5

Divide $1000 x^{3} + 4000 x^{2} + 5000 x - 541$ by $10 x - 1$ .

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Step 4.3.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$10 x$

$1$

$1000 x^{3}$

$4000 x^{2}$

$5000 x$

$541$

Step 4.3.1.5.2

Divide the highest order term in the dividend $1000 x^{3}$ by the highest order term in divisor $10 x$ .

					$100 x^{2}$
	$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$

Step 4.3.1.5.3

Multiply the new quotient term by the divisor.

				$100 x^{2}$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			+	$1000 x^{3}$	-	$100 x^{2}$

Step 4.3.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $1000 x^{3} - 100 x^{2}$

				$100 x^{2}$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$

Step 4.3.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$100 x^{2}$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$

Step 4.3.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$100 x^{2}$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$

Step 4.3.1.5.7

Divide the highest order term in the dividend $4100 x^{2}$ by the highest order term in divisor $10 x$ .

				$100 x^{2}$	+	$410 x$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$

Step 4.3.1.5.8

Multiply the new quotient term by the divisor.

				$100 x^{2}$	+	$410 x$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					+	$4100 x^{2}$	-	$410 x$

Step 4.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4100 x^{2} - 410 x$

				$100 x^{2}$	+	$410 x$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$

Step 4.3.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$100 x^{2}$	+	$410 x$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$
							+	$5410 x$

Step 4.3.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$100 x^{2}$	+	$410 x$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$
							+	$5410 x$	-	$541$

Step 4.3.1.5.12

Divide the highest order term in the dividend $5410 x$ by the highest order term in divisor $10 x$ .

				$100 x^{2}$	+	$410 x$	+	$541$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$
							+	$5410 x$	-	$541$

Step 4.3.1.5.13

Multiply the new quotient term by the divisor.

				$100 x^{2}$	+	$410 x$	+	$541$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$
							+	$5410 x$	-	$541$
							+	$5410 x$	-	$541$

Step 4.3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $5410 x - 541$

				$100 x^{2}$	+	$410 x$	+	$541$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$
							+	$5410 x$	-	$541$
							-	$5410 x$	+	$541$

Step 4.3.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$100 x^{2}$	+	$410 x$	+	$541$
$10 x$	-	$1$		$1000 x^{3}$	+	$4000 x^{2}$	+	$5000 x$	-	$541$
			-	$1000 x^{3}$	+	$100 x^{2}$
					+	$4100 x^{2}$	+	$5000 x$
					-	$4100 x^{2}$	+	$410 x$
							+	$5410 x$	-	$541$
							-	$5410 x$	+	$541$
										$0$

Step 4.3.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$100 x^{2} + 410 x + 541$

Step 4.3.1.6

Write $1000 x^{3} + 4000 x^{2} + 5000 x - 541$ as a set of factors.

$0.0005 ((10 x - 1) (100 x^{2} + 410 x + 541)) = 0$

Step 4.3.2

Remove unnecessary parentheses.

$0.0005 (10 x - 1) (100 x^{2} + 410 x + 541) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$10 x - 1 = 0$

$100 x^{2} + 410 x + 541 = 0$

Step 6

Set $10 x - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $10 x - 1$ equal to $0$ .

$10 x - 1 = 0$

Step 6.2

Solve $10 x - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$10 x = 1$

Step 6.2.2

Divide each term in $10 x = 1$ by $10$ and simplify.

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Step 6.2.2.1

Divide each term in $10 x = 1$ by $10$ .

$\frac{10 x}{10} = \frac{1}{10}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $10$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{1}{10}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{10}$

Step 7

Set $100 x^{2} + 410 x + 541$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $100 x^{2} + 410 x + 541$ equal to $0$ .

$100 x^{2} + 410 x + 541 = 0$

Step 7.2

Solve $100 x^{2} + 410 x + 541 = 0$ for $x$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

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

Step 7.2.2

Substitute the values $a = 100$ , $b = 410$ , and $c = 541$ into the quadratic formula and solve for $x$ .

$\frac{- 410 \pm \sqrt{410^{2} - 4 \cdot (100 \cdot 541)}}{2 \cdot 100}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

Raise $410$ to the power of $2$ .

$x = \frac{- 410 \pm \sqrt{168100 - 4 \cdot 100 \cdot 541}}{2 \cdot 100}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 100 \cdot 541$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $100$ .

$x = \frac{- 410 \pm \sqrt{168100 - 400 \cdot 541}}{2 \cdot 100}$

Step 7.2.3.1.2.2

Multiply $- 400$ by $541$ .

$x = \frac{- 410 \pm \sqrt{168100 - 216400}}{2 \cdot 100}$

Step 7.2.3.1.3

Subtract $216400$ from $168100$ .

$x = \frac{- 410 \pm \sqrt{- 48300}}{2 \cdot 100}$

Step 7.2.3.1.4

Rewrite $- 48300$ as $- 1 (48300)$ .

$x = \frac{- 410 \pm \sqrt{- 1 \cdot 48300}}{2 \cdot 100}$

Step 7.2.3.1.5

Rewrite $\sqrt{- 1 (48300)}$ as $\sqrt{- 1} \cdot \sqrt{48300}$ .

$x = \frac{- 410 \pm \sqrt{- 1} \cdot \sqrt{48300}}{2 \cdot 100}$

Step 7.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 410 \pm i \cdot \sqrt{48300}}{2 \cdot 100}$

Step 7.2.3.1.7

Rewrite $48300$ as $10^{2} \cdot 483$ .

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Step 7.2.3.1.7.1

Factor $100$ out of $48300$ .

$x = \frac{- 410 \pm i \cdot \sqrt{100 (483)}}{2 \cdot 100}$

Step 7.2.3.1.7.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 410 \pm i \cdot \sqrt{10^{2} \cdot 483}}{2 \cdot 100}$

Step 7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 410 \pm i \cdot (10 \sqrt{483})}{2 \cdot 100}$

Step 7.2.3.1.9

Move $10$ to the left of $i$ .

$x = \frac{- 410 \pm 10 i \sqrt{483}}{2 \cdot 100}$

Step 7.2.3.2

Multiply $2$ by $100$ .

$x = \frac{- 410 \pm 10 i \sqrt{483}}{200}$

Step 7.2.3.3

Simplify $\frac{- 410 \pm 10 i \sqrt{483}}{200}$ .

$x = \frac{- 41 \pm i \sqrt{483}}{20}$

Step 7.2.4

The final answer is the combination of both solutions.

$x = - \frac{41 - i \sqrt{483}}{20}, - \frac{41 + i \sqrt{483}}{20}$

Step 8

The final solution is all the values that make $0.0005 (10 x - 1) (100 x^{2} + 410 x + 541) = 0$ true.

$x = \frac{1}{10}, - \frac{41 - i \sqrt{483}}{20}, - \frac{41 + i \sqrt{483}}{20}$

Step 9