Find the Roots/Zeros Using the Rational Roots Test f(x)=-2x^3-19x^2-138x+74

$f (x) = - 2 x^{3} - 19 x^{2} - 138 x + 74$

Step 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.

$p = \pm 1, \pm 2, \pm 37, \pm 74$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 37, \pm \frac{37}{2}, \pm 74$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$- 2 {(\frac{1}{2})}^{3} - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{1}{2}$ is a root of the polynomial.

Simplify each term.

Apply the product rule to $\frac{1}{2}$ .

$- 2 \frac{1^{3}}{2^{3}} - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

One to any power is one.

$- 2 \frac{1}{2^{3}} - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Raise $2$ to the power of $3$ .

$- 2 (\frac{1}{8}) - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{1}{8} - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Factor $2$ out of $8$ .

$2 \cdot - 1 \frac{1}{2 \cdot 4} - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Cancel the common factor.

$2 \cdot - 1 \frac{1}{2 \cdot 4} - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Rewrite the expression.

$- 1 (\frac{1}{4}) - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

$- (\frac{1}{4}) - 19 {(\frac{1}{2})}^{2} - 138 (\frac{1}{2}) + 74$

Apply the product rule to $\frac{1}{2}$ .

$- \frac{1}{4} - 19 \frac{1^{2}}{2^{2}} - 138 (\frac{1}{2}) + 74$

One to any power is one.

$- \frac{1}{4} - 19 \frac{1}{2^{2}} - 138 (\frac{1}{2}) + 74$

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

$- \frac{1}{4} - 19 (\frac{1}{4}) - 138 (\frac{1}{2}) + 74$

Combine $- 19$ and $\frac{1}{4}$ .

$- \frac{1}{4} + \frac{- 19}{4} - 138 (\frac{1}{2}) + 74$

Move the negative in front of the fraction.

$- \frac{1}{4} - \frac{19}{4} - 138 (\frac{1}{2}) + 74$

Cancel the common factor of $2$ .

Factor $2$ out of $- 138$ .

$- \frac{1}{4} - \frac{19}{4} + 2 (- 69) \frac{1}{2} + 74$

Cancel the common factor.

$- \frac{1}{4} - \frac{19}{4} + 2 \cdot - 69 \frac{1}{2} + 74$

Rewrite the expression.

$- \frac{1}{4} - \frac{19}{4} - 69 + 74$

Combine fractions.

Combine the numerators over the common denominator.

$- 69 + 74 + \frac{- 1 - 19}{4}$

Subtract $19$ from $- 1$ .

$- 69 + 74 + \frac{- 20}{4}$

Find the common denominator.

Write $- 69$ as a fraction with denominator $1$ .

$\frac{- 69}{1} + 74 + \frac{- 20}{4}$

Multiply $\frac{- 69}{1}$ by $\frac{4}{4}$ .

$\frac{- 69}{1} \cdot \frac{4}{4} + 74 + \frac{- 20}{4}$

Multiply $\frac{- 69}{1}$ by $\frac{4}{4}$ .

$\frac{- 69 \cdot 4}{4} + 74 + \frac{- 20}{4}$

Write $74$ as a fraction with denominator $1$ .

$\frac{- 69 \cdot 4}{4} + \frac{74}{1} + \frac{- 20}{4}$

Multiply $\frac{74}{1}$ by $\frac{4}{4}$ .

$\frac{- 69 \cdot 4}{4} + \frac{74}{1} \cdot \frac{4}{4} + \frac{- 20}{4}$

Multiply $\frac{74}{1}$ by $\frac{4}{4}$ .

$\frac{- 69 \cdot 4}{4} + \frac{74 \cdot 4}{4} + \frac{- 20}{4}$

Combine the numerators over the common denominator.

$\frac{- 69 \cdot 4 + 74 \cdot 4 - 20}{4}$

Simplify each term.

Multiply $- 69$ by $4$ .

$\frac{- 276 + 74 \cdot 4 - 20}{4}$

Multiply $74$ by $4$ .

$\frac{- 276 + 296 - 20}{4}$

Simplify the expression.

Add $- 276$ and $296$ .

$\frac{20 - 20}{4}$

Subtract $20$ from $20$ .

$\frac{0}{4}$

Divide $0$ by $4$ .

$0$

Step 5

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

$\frac{- 2 x^{3} - 19 x^{2} - 138 x + 74}{x - \frac{1}{2}}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0			\xa0

The first number in the dividend $(- 2)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0			\xa0
	$- 2$

Multiply the newest entry in the result $(- 2)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 1)$ under the next term in the dividend $(- 19)$ .

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0	$- 1$		\xa0
	$- 2$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0	$- 1$		\xa0
	$- 2$	$- 20$

Multiply the newest entry in the result $(- 20)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 10)$ under the next term in the dividend $(- 138)$ .

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0	$- 1$	$- 10$	\xa0
	$- 2$	$- 20$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0	$- 1$	$- 10$	\xa0
	$- 2$	$- 20$	$- 148$

Multiply the newest entry in the result $(- 148)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 74)$ under the next term in the dividend $(74)$ .

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0	$- 1$	$- 10$	$- 74$
	$- 2$	$- 20$	$- 148$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{1}{2}$	$- 2$	$- 19$	$- 138$	$74$
	\xa0	$- 1$	$- 10$	$- 74$
	$- 2$	$- 20$	$- 148$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$- 2 x^{2} + - 20 x - 148$

Simplify the quotient polynomial.

$- 2 x^{2} - 20 x - 148$

Step 7

Factor $2$ out of $- 2 x^{2} - 20 x - 148$ .

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

$(x - \frac{1}{2}) (2 (- x^{2}) - 20 x - 148)$

Factor $2$ out of $- 20 x$ .

$(x - \frac{1}{2}) (2 (- x^{2}) + 2 (- 10 x) - 148)$

Factor $2$ out of $- 148$ .

$(x - \frac{1}{2}) (2 (- x^{2}) + 2 (- 10 x) + 2 (- 74))$

Factor $2$ out of $2 (- x^{2}) + 2 (- 10 x)$ .

$(x - \frac{1}{2}) (2 (- x^{2} - 10 x) + 2 (- 74))$

Factor $2$ out of $2 (- x^{2} - 10 x) + 2 (- 74)$ .

$(x - \frac{1}{2}) (2 (- x^{2} - 10 x - 74))$

Step 8

Factor $- 2 x^{3} - 19 x^{2} - 138 x + 74$ 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.

$p = \pm 1, \pm 74, \pm 2, \pm 37 q = \pm 1, \pm 2$

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

$\pm 1, \pm 0.5, \pm 74, \pm 37, \pm 2, \pm 18.5$

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.

$- 2 \cdot {0.5}^{3} - 19 \cdot {0.5}^{2} - 138 \cdot 0.5 + 74$

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

$- 2 \cdot 0.125 - 19 \cdot {0.5}^{2} - 138 \cdot 0.5 + 74$

Multiply $- 2$ by $0.125$ .

$- 0.25 - 19 \cdot {0.5}^{2} - 138 \cdot 0.5 + 74$

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

$- 0.25 - 19 \cdot 0.25 - 138 \cdot 0.5 + 74$

Multiply $- 19$ by $0.25$ .

$- 0.25 - 4.75 - 138 \cdot 0.5 + 74$

Subtract $4.75$ from $- 0.25$ .

$- 5 - 138 \cdot 0.5 + 74$

Multiply $- 138$ by $0.5$ .

$- 5 - 69 + 74$

Subtract $69$ from $- 5$ .

$- 74 + 74$

Add $- 74$ and $74$ .

$0$

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.

$\frac{- 2 x^{3} - 19 x^{2} - 138 x + 74}{2 x - 1}$

Divide $- 2 x^{3} - 19 x^{2} - 138 x + 74$ 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 $0$ .

$2 x$

$1$

$2 x^{3}$

$19 x^{2}$

$138 x$

$74$

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

				-	$x^{2}$
	$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			-	$2 x^{3}$	+	$x^{2}$

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

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$

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

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$

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

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$

Divide the highest order term in the dividend $- 20 x^{2}$ by the highest order term in divisor $2 x$ .

			-	$x^{2}$	-	$10 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$10 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					-	$20 x^{2}$	+	$10 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{2} + 10 x$

			-	$x^{2}$	-	$10 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$

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

			-	$x^{2}$	-	$10 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$
							-	$148 x$

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

			-	$x^{2}$	-	$10 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$
							-	$148 x$	+	$74$

Divide the highest order term in the dividend $- 148 x$ by the highest order term in divisor $2 x$ .

			-	$x^{2}$	-	$10 x$	-	$74$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$
							-	$148 x$	+	$74$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$10 x$	-	$74$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$
							-	$148 x$	+	$74$
							-	$148 x$	+	$74$

The expression needs to be subtracted from the dividend, so change all the signs in $- 148 x + 74$

			-	$x^{2}$	-	$10 x$	-	$74$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$
							-	$148 x$	+	$74$
							+	$148 x$	-	$74$

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

			-	$x^{2}$	-	$10 x$	-	$74$
$2 x$	-	$1$	-	$2 x^{3}$	-	$19 x^{2}$	-	$138 x$	+	$74$
			+	$2 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	-	$138 x$
					+	$20 x^{2}$	-	$10 x$
							-	$148 x$	+	$74$
							+	$148 x$	-	$74$
										$0$

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

$- x^{2} - 10 x - 74$

Write $- 2 x^{3} - 19 x^{2} - 138 x + 74$ as a set of factors.

$(2 x - 1) (- x^{2} - 10 x - 74) = 0$

Step 9

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

$2 x - 1 = 0$

$- x^{2} - 10 x - 74 = 0$

Step 10

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 $1$ .

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

Step 11

Set $- x^{2} - 10 x - 74$ equal to $0$ and solve for $x$ .

Set $- x^{2} - 10 x - 74$ equal to $0$ .

$- x^{2} - 10 x - 74 = 0$

Solve $- x^{2} - 10 x - 74 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = - 10$ , and $c = - 74$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 1 \cdot - 74)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 74}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 74$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 4 \cdot - 74}}{2 \cdot - 1}$

Multiply $4$ by $- 74$ .

$x = \frac{10 \pm \sqrt{100 - 296}}{2 \cdot - 1}$

Subtract $296$ from $100$ .

$x = \frac{10 \pm \sqrt{- 196}}{2 \cdot - 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 196}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot - 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{196}}{2 \cdot - 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{14^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 14}{2 \cdot - 1}$

Move $14$ to the left of $i$ .

$x = \frac{10 \pm 14 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{10 \pm 14 i}{- 2}$

Simplify $\frac{10 \pm 14 i}{- 2}$ .

$x = \frac{5 \pm 7 i}{- 1}$

Move the negative one from the denominator of $\frac{5 \pm 7 i}{- 1}$ .

$x = - 1 \cdot (5 \pm 7 i)$

Rewrite $- 1 \cdot (5 \pm 7 i)$ as $- (5 \pm 7 i)$ .

$x = - (5 \pm 7 i)$

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 74}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 74$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 4 \cdot - 74}}{2 \cdot - 1}$

Multiply $4$ by $- 74$ .

$x = \frac{10 \pm \sqrt{100 - 296}}{2 \cdot - 1}$

Subtract $296$ from $100$ .

$x = \frac{10 \pm \sqrt{- 196}}{2 \cdot - 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 196}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot - 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{196}}{2 \cdot - 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{14^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 14}{2 \cdot - 1}$

Move $14$ to the left of $i$ .

$x = \frac{10 \pm 14 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{10 \pm 14 i}{- 2}$

Simplify $\frac{10 \pm 14 i}{- 2}$ .

$x = \frac{5 \pm 7 i}{- 1}$

Move the negative one from the denominator of $\frac{5 \pm 7 i}{- 1}$ .

$x = - 1 \cdot (5 \pm 7 i)$

Rewrite $- 1 \cdot (5 \pm 7 i)$ as $- (5 \pm 7 i)$ .

$x = - (5 \pm 7 i)$

Change the $\pm$ to $+$ .

$x = - (5 + 7 i)$

Apply the distributive property.

$x = - 1 \cdot 5 - (7 i)$

Multiply $- 1$ by $5$ .

$x = - 5 - (7 i)$

Multiply $7$ by $- 1$ .

$x = - 5 - 7 i$

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 74}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 74$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 4 \cdot - 74}}{2 \cdot - 1}$

Multiply $4$ by $- 74$ .

$x = \frac{10 \pm \sqrt{100 - 296}}{2 \cdot - 1}$

Subtract $296$ from $100$ .

$x = \frac{10 \pm \sqrt{- 196}}{2 \cdot - 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 196}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot - 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{196}}{2 \cdot - 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{14^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 14}{2 \cdot - 1}$

Move $14$ to the left of $i$ .

$x = \frac{10 \pm 14 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{10 \pm 14 i}{- 2}$

Simplify $\frac{10 \pm 14 i}{- 2}$ .

$x = \frac{5 \pm 7 i}{- 1}$

Move the negative one from the denominator of $\frac{5 \pm 7 i}{- 1}$ .

$x = - 1 \cdot (5 \pm 7 i)$

Rewrite $- 1 \cdot (5 \pm 7 i)$ as $- (5 \pm 7 i)$ .

$x = - (5 \pm 7 i)$

Change the $\pm$ to $-$ .

$x = - (5 - 7 i)$

Apply the distributive property.

$x = - 1 \cdot 5 - (- 7 i)$

Multiply $- 1$ by $5$ .

$x = - 5 - (- 7 i)$

Multiply $- 7$ by $- 1$ .

$x = - 5 + 7 i$

The final answer is the combination of both solutions.

$x = - 5 - 7 i, - 5 + 7 i$

Step 12

The final solution is all the values that make $(2 x - 1) (- x^{2} - 10 x - 74) = 0$ true.

$x = \frac{1}{2}, - 5 - 7 i, - 5 + 7 i$

Step 13

Find the Roots/Zeros Using the Rational Roots Test f(x)=-2x^3-19x^2-138x+74

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