Calculus Examples

$1485 = x^{2} (0.08 x + 59)$

Step 1

Rewrite the equation as $x^{2} (0.08 x + 59) = 1485$ .

$x^{2} (0.08 x + 59) = 1485$

Step 2

Simplify $x^{2} (0.08 x + 59)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$x^{2} (0.08 x) + x^{2} \cdot 59 = 1485$

Step 2.1.2

Reorder.

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Step 2.1.2.1

Rewrite using the commutative property of multiplication.

$0.08 x^{2} x + x^{2} \cdot 59 = 1485$

Step 2.1.2.2

Move $59$ to the left of $x^{2}$ .

$0.08 x^{2} x + 59 \cdot x^{2} = 1485$

Step 2.2

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

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Step 2.2.1

Move $x$ .

$0.08 (x \cdot x^{2}) + 59 \cdot x^{2} = 1485$

Step 2.2.2

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

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Step 2.2.2.1

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

$0.08 (x^{1} x^{2}) + 59 \cdot x^{2} = 1485$

Step 2.2.2.2

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

$0.08 x^{1 + 2} + 59 \cdot x^{2} = 1485$

Step 2.2.3

Add $1$ and $2$ .

$0.08 x^{3} + 59 \cdot x^{2} = 1485$

$0.08 x^{3} + 59 x^{2} = 1485$

Step 3

Subtract $1485$ from both sides of the equation.

$0.08 x^{3} + 59 x^{2} - 1485 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $0.04$ out of $0.08 x^{3} + 59 x^{2} - 1485$ .

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Step 4.1.1

Factor $0.04$ out of $0.08 x^{3}$ .

$0.04 (2 x^{3}) + 59 x^{2} - 1485 = 0$

Step 4.1.2

Factor $0.04$ out of $59 x^{2}$ .

$0.04 (2 x^{3}) + 0.04 (1475 x^{2}) - 1485 = 0$

Step 4.1.3

Factor $0.04$ out of $- 1485$ .

$0.04 (2 x^{3}) + 0.04 (1475 x^{2}) + 0.04 (- 37125) = 0$

Step 4.1.4

Factor $0.04$ out of $0.04 (2 x^{3}) + 0.04 (1475 x^{2})$ .

$0.04 (2 x^{3} + 1475 x^{2}) + 0.04 (- 37125) = 0$

Step 4.1.5

Factor $0.04$ out of $0.04 (2 x^{3} + 1475 x^{2}) + 0.04 (- 37125)$ .

$0.04 (2 x^{3} + 1475 x^{2} - 37125) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $2 x^{3} + 1475 x^{2} - 37125$ using the rational roots test.

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Step 4.2.1.1

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.

$p = \pm 1, \pm 37125, \pm 3, \pm 12375, \pm 5, \pm 7425, \pm 9, \pm 4125, \pm 11, \pm 3375, \pm 15, \pm 2475, \pm 25, \pm 1485, \pm 27, \pm 1375, \pm 33, \pm 1125, \pm 45, \pm 825, \pm 55, \pm 675, \pm 75, \pm 495, \pm 99, \pm 375, \pm 125, \pm 297, \pm 135, \pm 275, \pm 165, \pm 225$

$q = \pm 1, \pm 2$

Step 4.2.1.2

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

$\pm 1, \pm 0.5, \pm 37125, \pm 18562.5, \pm 3, \pm 1.5, \pm 12375, \pm 6187.5, \pm 5, \pm 2.5, \pm 7425, \pm 3712.5, \pm 9, \pm 4.5, \pm 4125, \pm 2062.5, \pm 11, \pm 5.5, \pm 3375, \pm 1687.5, \pm 15, \pm 7.5, \pm 2475, \pm 1237.5, \pm 25, \pm 12.5, \pm 1485, \pm 742.5, \pm 27, \pm 13.5, \pm 1375, \pm 687.5, \pm 33, \pm 16.5, \pm 1125, \pm 562.5, \pm 45, \pm 22.5, \pm 825, \pm 412.5, \pm 55, \pm 27.5, \pm 675, \pm 337.5, \pm 75, \pm 37.5, \pm 495, \pm 247.5, \pm 99, \pm 49.5, \pm 375, \pm 187.5, \pm 125, \pm 62.5, \pm 297, \pm 148.5, \pm 135, \pm 67.5, \pm 275, \pm 137.5, \pm 165, \pm 82.5, \pm 225, \pm 112.5$

Step 4.2.1.3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

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Step 4.2.1.3.1

Substitute $5$ into the polynomial.

$2 \cdot 5^{3} + 1475 \cdot 5^{2} - 37125$

Step 4.2.1.3.2

Raise $5$ to the power of $3$ .

$2 \cdot 125 + 1475 \cdot 5^{2} - 37125$

Step 4.2.1.3.3

Multiply $2$ by $125$ .

$250 + 1475 \cdot 5^{2} - 37125$

Step 4.2.1.3.4

Raise $5$ to the power of $2$ .

$250 + 1475 \cdot 25 - 37125$

Step 4.2.1.3.5

Multiply $1475$ by $25$ .

$250 + 36875 - 37125$

Step 4.2.1.3.6

Add $250$ and $36875$ .

$37125 - 37125$

Step 4.2.1.3.7

Subtract $37125$ from $37125$ .

$0$

Step 4.2.1.4

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 1475 x^{2} - 37125}{x - 5}$

Step 4.2.1.5

Divide $2 x^{3} + 1475 x^{2} - 37125$ by $x - 5$ .

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Step 4.2.1.5.1

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

$x$

$5$

$2 x^{3}$

$1475 x^{2}$

$0 x$

$37125$

Step 4.2.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$

Step 4.2.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			+	$2 x^{3}$	-	$10 x^{2}$

Step 4.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 10 x^{2}$

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$

Step 4.2.1.5.5

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

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$

Step 4.2.1.5.6

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

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$

Step 4.2.1.5.7

Divide the highest order term in the dividend $1485 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$1485 x$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$

Step 4.2.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$1485 x$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					+	$1485 x^{2}$	-	$7425 x$

Step 4.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $1485 x^{2} - 7425 x$

				$2 x^{2}$	+	$1485 x$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$

Step 4.2.1.5.10

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

				$2 x^{2}$	+	$1485 x$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$
							+	$7425 x$

Step 4.2.1.5.11

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

				$2 x^{2}$	+	$1485 x$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$
							+	$7425 x$	-	$37125$

Step 4.2.1.5.12

Divide the highest order term in the dividend $7425 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$1485 x$	+	$7425$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$
							+	$7425 x$	-	$37125$

Step 4.2.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$1485 x$	+	$7425$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$
							+	$7425 x$	-	$37125$
							+	$7425 x$	-	$37125$

Step 4.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $7425 x - 37125$

				$2 x^{2}$	+	$1485 x$	+	$7425$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$
							+	$7425 x$	-	$37125$
							-	$7425 x$	+	$37125$

Step 4.2.1.5.15

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

				$2 x^{2}$	+	$1485 x$	+	$7425$
$x$	-	$5$		$2 x^{3}$	+	$1475 x^{2}$	+	$0 x$	-	$37125$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$1485 x^{2}$	+	$0 x$
					-	$1485 x^{2}$	+	$7425 x$
							+	$7425 x$	-	$37125$
							-	$7425 x$	+	$37125$
										$0$

Step 4.2.1.5.16

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

$2 x^{2} + 1485 x + 7425$

Step 4.2.1.6

Write $2 x^{3} + 1475 x^{2} - 37125$ as a set of factors.

$0.04 ((x - 5) (2 x^{2} + 1485 x + 7425)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$0.04 (x - 5) (2 x^{2} + 1485 x + 7425) = 0$

Step 5

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

$x - 5 = 0$

$2 x^{2} + 1485 x + 7425 = 0$

Step 6

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$x = 5$

Step 7

Set $2 x^{2} + 1485 x + 7425$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x^{2} + 1485 x + 7425$ equal to $0$ .

$2 x^{2} + 1485 x + 7425 = 0$

Step 7.2

Solve $2 x^{2} + 1485 x + 7425 = 0$ 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 $a = 2$ , $b = 1485$ , and $c = 7425$ into the quadratic formula and solve for $x$ .

$\frac{- 1485 \pm \sqrt{1485^{2} - 4 \cdot (2 \cdot 7425)}}{2 \cdot 2}$

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 $1485$ to the power of $2$ .

$x = \frac{- 1485 \pm \sqrt{2205225 - 4 \cdot 2 \cdot 7425}}{2 \cdot 2}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 7425$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1485 \pm \sqrt{2205225 - 8 \cdot 7425}}{2 \cdot 2}$

Step 7.2.3.1.2.2

Multiply $- 8$ by $7425$ .

$x = \frac{- 1485 \pm \sqrt{2205225 - 59400}}{2 \cdot 2}$

Step 7.2.3.1.3

Subtract $59400$ from $2205225$ .

$x = \frac{- 1485 \pm \sqrt{2145825}}{2 \cdot 2}$

Step 7.2.3.1.4

Rewrite $2145825$ as $255^{2} \cdot 33$ .

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Step 7.2.3.1.4.1

Factor $65025$ out of $2145825$ .

$x = \frac{- 1485 \pm \sqrt{65025 (33)}}{2 \cdot 2}$

Step 7.2.3.1.4.2

Rewrite $65025$ as $255^{2}$ .

$x = \frac{- 1485 \pm \sqrt{255^{2} \cdot 33}}{2 \cdot 2}$

Step 7.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 1485 \pm 255 \sqrt{33}}{2 \cdot 2}$

Step 7.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 1485 \pm 255 \sqrt{33}}{4}$

Step 7.2.4

The final answer is the combination of both solutions.

$x = - \frac{1485 - 255 \sqrt{33}}{4}, - \frac{1485 + 255 \sqrt{33}}{4}$

Step 8

The final solution is all the values that make $0.04 (x - 5) (2 x^{2} + 1485 x + 7425) = 0$ true.

$x = 5, - \frac{1485 - 255 \sqrt{33}}{4}, - \frac{1485 + 255 \sqrt{33}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 5, - \frac{1485 - 255 \sqrt{33}}{4}, - \frac{1485 + 255 \sqrt{33}}{4}$

Decimal Form:

$x = 5, - 5.03413128 \dots, - 737.46586871 \dots$