Solve for x 2x^2+17x+27=18/x

$2 x^{2} + 17 x + 27 = \frac{18}{x}$

Step 1

Find the LCD of the terms in the equation.

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

$1, 1, 1, x$

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $2 x^{2} + 17 x + 27 = \frac{18}{x}$ by $x$ to eliminate the fractions.

Multiply each term in $2 x^{2} + 17 x + 27 = \frac{18}{x}$ by $x$ .

$2 x^{2} x + 17 x \cdot x + 27 x = \frac{18}{x} x$

Simplify the left side.

Simplify each term.

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

Move $x$ .

$2 (x \cdot x^{2}) + 17 x \cdot x + 27 x = \frac{18}{x} x$

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

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

$2 (x^{1} x^{2}) + 17 x \cdot x + 27 x = \frac{18}{x} x$

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

$2 x^{1 + 2} + 17 x \cdot x + 27 x = \frac{18}{x} x$

Add $1$ and $2$ .

$2 x^{3} + 17 x \cdot x + 27 x = \frac{18}{x} x$

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

Move $x$ .

$2 x^{3} + 17 (x \cdot x) + 27 x = \frac{18}{x} x$

Multiply $x$ by $x$ .

$2 x^{3} + 17 x^{2} + 27 x = \frac{18}{x} x$

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

$2 x^{3} + 17 x^{2} + 27 x = \frac{18}{x} x$

Rewrite the expression.

$2 x^{3} + 17 x^{2} + 27 x = 18$

Step 3

Solve the equation.

Subtract $18$ from both sides of the equation.

$2 x^{3} + 17 x^{2} + 27 x - 18 = 0$

Factor the left side of the equation.

Factor $2 x^{3} + 17 x^{2} + 27 x - 18$ using the rational roots test.

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.

$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6 q = \pm 1, \pm 2$

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

$\pm 1, \pm 0.5, \pm 18, \pm 9, \pm 2, \pm 4.5, \pm 3, \pm 1.5, \pm 6$

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} + 17 \cdot {0.5}^{2} + 27 \cdot 0.5 - 18$

Raise $0.5$ to the power of $3$ .

$2 \cdot 0.125 + 17 \cdot {0.5}^{2} + 27 \cdot 0.5 - 18$

Multiply $2$ by $0.125$ .

$0.25 + 17 \cdot {0.5}^{2} + 27 \cdot 0.5 - 18$

Raise $0.5$ to the power of $2$ .

$0.25 + 17 \cdot 0.25 + 27 \cdot 0.5 - 18$

Multiply $17$ by $0.25$ .

$0.25 + 4.25 + 27 \cdot 0.5 - 18$

Add $0.25$ and $4.25$ .

$4.5 + 27 \cdot 0.5 - 18$

Multiply $27$ by $0.5$ .

$4.5 + 13.5 - 18$

Add $4.5$ and $13.5$ .

$18 - 18$

Subtract $18$ from $18$ .

$0$

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 17 x^{2} + 27 x - 18}{2 x - 1}$

Divide $2 x^{3} + 17 x^{2} + 27 x - 18$ by $2 x - 1$ .

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

$2 x$

$1$

$2 x^{3}$

$17 x^{2}$

$27 x$

$18$

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

					$x^{2}$
	$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			+	$2 x^{3}$	-	$x^{2}$

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$

Divide the highest order term in the dividend $18 x^{2}$ by the highest order term in divisor $2 x$ .

				$x^{2}$	+	$9 x$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					+	$18 x^{2}$	-	$9 x$

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{2} - 9 x$

				$x^{2}$	+	$9 x$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$

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

				$x^{2}$	+	$9 x$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$
							+	$36 x$

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

				$x^{2}$	+	$9 x$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$
							+	$36 x$	-	$18$

Divide the highest order term in the dividend $36 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	+	$9 x$	+	$18$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$
							+	$36 x$	-	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$	+	$18$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$
							+	$36 x$	-	$18$
							+	$36 x$	-	$18$

The expression needs to be subtracted from the dividend, so change all the signs in $36 x - 18$

				$x^{2}$	+	$9 x$	+	$18$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$
							+	$36 x$	-	$18$
							-	$36 x$	+	$18$

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

				$x^{2}$	+	$9 x$	+	$18$
$2 x$	-	$1$		$2 x^{3}$	+	$17 x^{2}$	+	$27 x$	-	$18$
			-	$2 x^{3}$	+	$x^{2}$
					+	$18 x^{2}$	+	$27 x$
					-	$18 x^{2}$	+	$9 x$
							+	$36 x$	-	$18$
							-	$36 x$	+	$18$
										$0$

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

$x^{2} + 9 x + 18$

Write $2 x^{3} + 17 x^{2} + 27 x - 18$ as a set of factors.

$(2 x - 1) (x^{2} + 9 x + 18) = 0$

Factor $x^{2} + 9 x + 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$(2 x - 1) ((x + 3) (x + 6)) = 0$

Remove unnecessary parentheses.

$(2 x - 1) (x + 3) (x + 6) = 0$

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

$2 x - 1 = 0$

$x + 3 = 0$

$x + 6 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(2 x - 1) (x + 3) (x + 6) = 0$ true.

$x = \frac{1}{2}, - 3, - 6$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, - 3, - 6$

Decimal Form:

$x = 0.5, - 3, - 6$

Solve for x 2x^2+17x+27=18/x

Related Questions

Support LCMGCF.com by disabling your adblocker.