Finite Math Examples

$4 x^{3} - 14 x^{2} + 25$

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.

$p = \pm 1, \pm 5, \pm 25$

$q = \pm 1, \pm 2, \pm 4$

Step 2

Find every combination of

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 25, \pm \frac{25}{2}, \pm \frac{25}{4}$

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.

$4 {(\frac{5}{2})}^{3} - 14 {(\frac{5}{2})}^{2} + 25$

Step 4

Simplify the expression. In this case, the expression is equal to

$0$ so

$x = \frac{5}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to

$\frac{5}{2}$ .

$4 \frac{5^{3}}{2^{3}} - 14 {(\frac{5}{2})}^{2} + 25$

Step 4.1.2

Raise

$5$ to the power of

$3$ .

$4 \frac{125}{2^{3}} - 14 {(\frac{5}{2})}^{2} + 25$

Step 4.1.3

Raise

$2$ to the power of

$3$ .

$4 (\frac{125}{8}) - 14 {(\frac{5}{2})}^{2} + 25$

Step 4.1.4

Cancel the common factor of

$4$ .

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Step 4.1.4.1

Factor

$4$ out of

$8$ .

$4 \frac{125}{4 (2)} - 14 {(\frac{5}{2})}^{2} + 25$

Step 4.1.4.2

Cancel the common factor.

$4 \frac{125}{4 \cdot 2} - 14 {(\frac{5}{2})}^{2} + 25$

Step 4.1.4.3

Rewrite the expression.

$\frac{125}{2} - 14 {(\frac{5}{2})}^{2} + 25$

Step 4.1.5

Apply the product rule to

$\frac{5}{2}$ .

$\frac{125}{2} - 14 \frac{5^{2}}{2^{2}} + 25$

Step 4.1.6

Raise

$5$ to the power of

$2$ .

$\frac{125}{2} - 14 \frac{25}{2^{2}} + 25$

Step 4.1.7

Raise

$2$ to the power of

$2$ .

$\frac{125}{2} - 14 (\frac{25}{4}) + 25$

Step 4.1.8

Cancel the common factor of

$2$ .

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Step 4.1.8.1

Factor

$2$ out of

$- 14$ .

$\frac{125}{2} + 2 (- 7) \frac{25}{4} + 25$

Step 4.1.8.2

Factor

$2$ out of

$4$ .

$\frac{125}{2} + 2 \cdot - 7 \frac{25}{2 \cdot 2} + 25$

Step 4.1.8.3

Cancel the common factor.

$\frac{125}{2} + 2 \cdot - 7 \frac{25}{2 \cdot 2} + 25$

Step 4.1.8.4

Rewrite the expression.

$\frac{125}{2} - 7 (\frac{25}{2}) + 25$

Step 4.1.9

Combine

$- 7$ and

$\frac{25}{2}$ .

$\frac{125}{2} + \frac{- 7 \cdot 25}{2} + 25$

Step 4.1.10

Multiply

$- 7$ by

$25$ .

$\frac{125}{2} + \frac{- 175}{2} + 25$

Step 4.1.11

Move the negative in front of the fraction.

$\frac{125}{2} - \frac{175}{2} + 25$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$25 + \frac{125 - 175}{2}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Subtract

$175$ from

$125$ .

$25 + \frac{- 50}{2}$

Step 4.2.2.2

Divide

$- 50$ by

$2$ .

$25 - 25$

Step 4.2.2.3

Subtract

$25$ from

$25$ .

$0$

Step 5

Since

$\frac{5}{2}$ is a known root, divide the polynomial by

$x - \frac{5}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 14 x^{2} + 25}{x - \frac{5}{2}}$

Step 6

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

$1$ .

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Step 6.1

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

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$

Step 6.2

The first number in the dividend

$(4)$ is put into the first position of the result area (below the horizontal line).

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$

	$4$

Step 6.3

Multiply the newest entry in the result

$(4)$ by the divisor

$(\frac{5}{2})$ and place the result of

$(10)$ under the next term in the dividend

$(- 14)$ .

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$
		$10$
	$4$

Step 6.4

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

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$
		$10$
	$4$	$- 4$

Step 6.5

Multiply the newest entry in the result

$(- 4)$ by the divisor

$(\frac{5}{2})$ and place the result of

$(- 10)$ under the next term in the dividend

$(0)$ .

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$
		$10$	$- 10$
	$4$	$- 4$

Step 6.6

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

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$
		$10$	$- 10$
	$4$	$- 4$	$- 10$

Step 6.7

Multiply the newest entry in the result

$(- 10)$ by the divisor

$(\frac{5}{2})$ and place the result of

$(- 25)$ under the next term in the dividend

$(25)$ .

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$
		$10$	$- 10$	$- 25$
	$4$	$- 4$	$- 10$

Step 6.8

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

$\frac{5}{2}$	$4$	$- 14$	$0$	$25$
		$10$	$- 10$	$- 25$
	$4$	$- 4$	$- 10$	$0$

Step 6.9

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

$4 x^{2} + - 4 x - 10$

Step 6.10

Simplify the quotient polynomial.

$4 x^{2} - 4 x - 10$

Step 7

Factor

$2$ out of

$4 x^{2} - 4 x - 10$ .

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Step 7.1

Factor

$2$ out of

$4 x^{2}$ .

$(x - \frac{5}{2}) (2 (2 x^{2}) - 4 x - 10)$

Step 7.2

Factor

$2$ out of

$- 4 x$ .

$(x - \frac{5}{2}) (2 (2 x^{2}) + 2 (- 2 x) - 10)$

Step 7.3

Factor

$2$ out of

$- 10$ .

$(x - \frac{5}{2}) (2 (2 x^{2}) + 2 (- 2 x) + 2 (- 5))$

Step 7.4

Factor

$2$ out of

$2 (2 x^{2}) + 2 (- 2 x)$ .

$(x - \frac{5}{2}) (2 (2 x^{2} - 2 x) + 2 (- 5))$

Step 7.5

Factor

$2$ out of

$2 (2 x^{2} - 2 x) + 2 (- 5)$ .

$(x - \frac{5}{2}) (2 (2 x^{2} - 2 x - 5))$

Step 8

Factor

$4 x^{3} - 14 x^{2} + 25$ using the rational roots test.

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Step 8.1

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.

$p = \pm 1, \pm 25, \pm 5$

$q = \pm 1, \pm 4, \pm 2$

Step 8.2

Find every combination of

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 25, \pm 6.25, \pm 12.5, \pm 5, \pm 1.25, \pm 2.5$

Step 8.3

Substitute

$2.5$ and simplify the expression. In this case, the expression is equal to

$0$ so

$2.5$ is a root of the polynomial.

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Step 8.3.1

Substitute

$2.5$ into the polynomial.

$4 \cdot {2.5}^{3} - 14 \cdot {2.5}^{2} + 25$

Step 8.3.2

Raise

$2.5$ to the power of

$3$ .

$4 \cdot 15.625 - 14 \cdot {2.5}^{2} + 25$

Step 8.3.3

Multiply

$4$ by

$15.625$ .

$62.5 - 14 \cdot {2.5}^{2} + 25$

Step 8.3.4

Raise

$2.5$ to the power of

$2$ .

$62.5 - 14 \cdot 6.25 + 25$

Step 8.3.5

Multiply

$- 14$ by

$6.25$ .

$62.5 - 87.5 + 25$

Step 8.3.6

Subtract

$87.5$ from

$62.5$ .

$- 25 + 25$

Step 8.3.7

Add

$- 25$ and

$25$ .

$0$

Step 8.4

Since

$2.5$ is a known root, divide the polynomial by

$2 x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 14 x^{2} + 25}{2 x - 5}$

Step 8.5

Divide

$4 x^{3} - 14 x^{2} + 25$ by

$2 x - 5$ .

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Step 8.5.1

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

$0$ .

$2 x$

$5$

$4 x^{3}$

$14 x^{2}$

$0 x$

$25$

Step 8.5.2

Divide the highest order term in the dividend

$4 x^{3}$ by the highest order term in divisor

$2 x$ .

					$2 x^{2}$
	$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$

Step 8.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			+	$4 x^{3}$	-	$10 x^{2}$

Step 8.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

$4 x^{3} - 10 x^{2}$

				$2 x^{2}$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$

Step 8.5.5

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

				$2 x^{2}$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$

Step 8.5.6

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

				$2 x^{2}$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 8.5.7

Divide the highest order term in the dividend

$- 4 x^{2}$ by the highest order term in divisor

$2 x$ .

				$2 x^{2}$	-	$2 x$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 8.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$

Step 8.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

$- 4 x^{2} + 10 x$

				$2 x^{2}$	-	$2 x$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$

Step 8.5.10

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

				$2 x^{2}$	-	$2 x$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$
							-	$10 x$

Step 8.5.11

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

				$2 x^{2}$	-	$2 x$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$
							-	$10 x$	+	$25$

Step 8.5.12

Divide the highest order term in the dividend

$- 10 x$ by the highest order term in divisor

$2 x$ .

				$2 x^{2}$	-	$2 x$	-	$5$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$
							-	$10 x$	+	$25$

Step 8.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$	-	$5$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$
							-	$10 x$	+	$25$
							-	$10 x$	+	$25$

Step 8.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$- 10 x + 25$

				$2 x^{2}$	-	$2 x$	-	$5$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$
							-	$10 x$	+	$25$
							+	$10 x$	-	$25$

Step 8.5.15

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

				$2 x^{2}$	-	$2 x$	-	$5$
$2 x$	-	$5$		$4 x^{3}$	-	$14 x^{2}$	+	$0 x$	+	$25$
			-	$4 x^{3}$	+	$10 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$
							-	$10 x$	+	$25$
							+	$10 x$	-	$25$
										$0$

Step 8.5.16

Since the remander is

$0$ , the final answer is the quotient.

$2 x^{2} - 2 x - 5$

Step 8.6

Write

$4 x^{3} - 14 x^{2} + 25$ as a set of factors.

$(2 x - 5) (2 x^{2} - 2 x - 5) = 0$

Step 9

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$2 x - 5 = 0$

$2 x^{2} - 2 x - 5 = 0$

Step 10

Set

$2 x - 5$ equal to

$0$ and solve for

$x$ .

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Step 10.1

Set

$2 x - 5$ equal to

$0$ .

$2 x - 5 = 0$

Step 10.2

Solve

$2 x - 5 = 0$ for

$x$ .

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Step 10.2.1

Add

$5$ to both sides of the equation.

$2 x = 5$

Step 10.2.2

Divide each term in

$2 x = 5$ by

$2$ and simplify.

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Step 10.2.2.1

Divide each term in

$2 x = 5$ by

$2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of

$2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 10.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{5}{2}$

Step 11

Set

$2 x^{2} - 2 x - 5$ equal to

$0$ and solve for

$x$ .

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Step 11.1

Set

$2 x^{2} - 2 x - 5$ equal to

$0$ .

$2 x^{2} - 2 x - 5 = 0$

Step 11.2

Solve

$2 x^{2} - 2 x - 5 = 0$ for

$x$ .

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Step 11.2.1

Use the quadratic formula to find the solutions.

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

Step 11.2.2

Substitute the values

$a = 2$ ,

$b = - 2$ , and

$c = - 5$ into the quadratic formula and solve for

$x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot - 5)}}{2 \cdot 2}$

Step 11.2.3

Simplify.

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Step 11.2.3.1

Simplify the numerator.

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Step 11.2.3.1.1

Raise

$- 2$ to the power of

$2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 11.2.3.1.2

Multiply

$- 4 \cdot 2 \cdot - 5$ .

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Step 11.2.3.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 5}}{2 \cdot 2}$

Step 11.2.3.1.2.2

Multiply

$- 8$ by

$- 5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot 2}$

Step 11.2.3.1.3

Add

$4$ and

$40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot 2}$

Step 11.2.3.1.4

Rewrite

$44$ as

$2^{2} \cdot 11$ .

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Step 11.2.3.1.4.1

Factor

$4$ out of

$44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Step 11.2.3.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Step 11.2.3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot 2}$

Step 11.2.3.2

Multiply

$2$ by

$2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{4}$

Step 11.2.3.3

Simplify

$\frac{2 \pm 2 \sqrt{11}}{4}$ .

$x = \frac{1 \pm \sqrt{11}}{2}$

Step 11.2.4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 11.2.4.1

Simplify the numerator.

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Step 11.2.4.1.1

Raise

$- 2$ to the power of

$2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 11.2.4.1.2

Multiply

$- 4 \cdot 2 \cdot - 5$ .

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Step 11.2.4.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 5}}{2 \cdot 2}$

Step 11.2.4.1.2.2

Multiply

$- 8$ by

$- 5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot 2}$

Step 11.2.4.1.3

Add

$4$ and

$40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot 2}$

Step 11.2.4.1.4

Rewrite

$44$ as

$2^{2} \cdot 11$ .

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Step 11.2.4.1.4.1

Factor

$4$ out of

$44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Step 11.2.4.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Step 11.2.4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot 2}$

Step 11.2.4.2

Multiply

$2$ by

$2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{4}$

Step 11.2.4.3

Simplify

$\frac{2 \pm 2 \sqrt{11}}{4}$ .

$x = \frac{1 \pm \sqrt{11}}{2}$

Step 11.2.4.4

Change the

$\pm$ to

$+$ .

$x = \frac{1 + \sqrt{11}}{2}$

Step 11.2.5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 11.2.5.1

Simplify the numerator.

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Step 11.2.5.1.1

Raise

$- 2$ to the power of

$2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 11.2.5.1.2

Multiply

$- 4 \cdot 2 \cdot - 5$ .

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Step 11.2.5.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 5}}{2 \cdot 2}$

Step 11.2.5.1.2.2

Multiply

$- 8$ by

$- 5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot 2}$

Step 11.2.5.1.3

Add

$4$ and

$40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot 2}$

Step 11.2.5.1.4

Rewrite

$44$ as

$2^{2} \cdot 11$ .

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Step 11.2.5.1.4.1

Factor

$4$ out of

$44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Step 11.2.5.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Step 11.2.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot 2}$

Step 11.2.5.2

Multiply

$2$ by

$2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{4}$

Step 11.2.5.3

Simplify

$\frac{2 \pm 2 \sqrt{11}}{4}$ .

$x = \frac{1 \pm \sqrt{11}}{2}$

Step 11.2.5.4

Change the

$\pm$ to

$-$ .

$x = \frac{1 - \sqrt{11}}{2}$

Step 11.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{11}}{2}, \frac{1 - \sqrt{11}}{2}$

Step 12

The final solution is all the values that make

$(2 x - 5) (2 x^{2} - 2 x - 5) = 0$ true.

$x = \frac{5}{2}, \frac{1 + \sqrt{11}}{2}, \frac{1 - \sqrt{11}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}, \frac{1 + \sqrt{11}}{2}, \frac{1 - \sqrt{11}}{2}$

Decimal Form:

$x = 2.5, 2.15831239 \dots, - 1.15831239 \dots$

Mixed Number Form:

$x = 2 \frac{1}{2}, \frac{1 + \sqrt{11}}{2}, \frac{1 - \sqrt{11}}{2}$

Step 14