Calculus Examples

$f (x) = 2 x^{5} + 5 x^{4} + 49 x^{3} + 119 x^{2} - 25 x - 150$

Step 1

Set $2 x^{5} + 5 x^{4} + 49 x^{3} + 119 x^{2} - 25 x - 150$ equal to $0$ .

$2 x^{5} + 5 x^{4} + 49 x^{3} + 119 x^{2} - 25 x - 150 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$2 x^{5} + 49 x^{3} - 25 x + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.2

Factor $x$ out of $2 x^{5} + 49 x^{3} - 25 x$ .

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Step 2.1.2.1

Factor $x$ out of $2 x^{5}$ .

$x (2 x^{4}) + 49 x^{3} - 25 x + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.2.2

Factor $x$ out of $49 x^{3}$ .

$x (2 x^{4}) + x (49 x^{2}) - 25 x + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.2.3

Factor $x$ out of $- 25 x$ .

$x (2 x^{4}) + x (49 x^{2}) + x \cdot - 25 + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.2.4

Factor $x$ out of $x (2 x^{4}) + x (49 x^{2})$ .

$x (2 x^{4} + 49 x^{2}) + x \cdot - 25 + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.2.5

Factor $x$ out of $x (2 x^{4} + 49 x^{2}) + x \cdot - 25$ .

$x (2 x^{4} + 49 x^{2} - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (2 {(x^{2})}^{2} + 49 x^{2} - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (2 u^{2} + 49 u - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.5

Factor by grouping.

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Step 2.1.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 25 = - 50$ and whose sum is $b = 49$ .

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Step 2.1.5.1.1

Factor $49$ out of $49 u$ .

$x (2 u^{2} + 49 (u) - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.5.1.2

Rewrite $49$ as $- 1$ plus $50$

$x (2 u^{2} + (- 1 + 50) u - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.5.1.3

Apply the distributive property.

$x (2 u^{2} - 1 u + 50 u - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.5.2

Factor out the greatest common factor from each group.

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Step 2.1.5.2.1

Group the first two terms and the last two terms.

$x ((2 u^{2} - 1 u) + 50 u - 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.5.2.2

Factor out the greatest common factor (GCF) from each group.

$x (u (2 u - 1) + 25 (2 u - 1)) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.5.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$x ((2 u - 1) (u + 25)) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.6

Factor.

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Step 2.1.6.1

Replace all occurrences of $u$ with $x^{2}$ .

$x ((2 x^{2} - 1) (x^{2} + 25)) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.6.2

Remove unnecessary parentheses.

$x (2 x^{2} - 1) (x^{2} + 25) + 5 x^{4} + 119 x^{2} - 150 = 0$

Step 2.1.7

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (2 x^{2} - 1) (x^{2} + 25) + 5 {(x^{2})}^{2} + 119 x^{2} - 150 = 0$

Step 2.1.8

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (2 x^{2} - 1) (x^{2} + 25) + 5 u^{2} + 119 u - 150 = 0$

Step 2.1.9

Factor by grouping.

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Step 2.1.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 150 = - 750$ and whose sum is $b = 119$ .

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Step 2.1.9.1.1

Factor $119$ out of $119 u$ .

$x (2 x^{2} - 1) (x^{2} + 25) + 5 u^{2} + 119 (u) - 150 = 0$

Step 2.1.9.1.2

Rewrite $119$ as $- 6$ plus $125$

$x (2 x^{2} - 1) (x^{2} + 25) + 5 u^{2} + (- 6 + 125) u - 150 = 0$

Step 2.1.9.1.3

Apply the distributive property.

$x (2 x^{2} - 1) (x^{2} + 25) + 5 u^{2} - 6 u + 125 u - 150 = 0$

Step 2.1.9.2

Factor out the greatest common factor from each group.

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Step 2.1.9.2.1

Group the first two terms and the last two terms.

$x (2 x^{2} - 1) (x^{2} + 25) + 5 u^{2} - 6 u + 125 u - 150 = 0$

Step 2.1.9.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x^{2} - 1) (x^{2} + 25) + u (5 u - 6) + 25 (5 u - 6) = 0$

Step 2.1.9.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 6$ .

$x (2 x^{2} - 1) (x^{2} + 25) + (5 u - 6) (u + 25) = 0$

Step 2.1.10

Replace all occurrences of $u$ with $x^{2}$ .

$x (2 x^{2} - 1) (x^{2} + 25) + (5 x^{2} - 6) (x^{2} + 25) = 0$

Step 2.1.11

Factor $x^{2} + 25$ out of $x (2 x^{2} - 1) (x^{2} + 25) + (5 x^{2} - 6) (x^{2} + 25)$ .

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Step 2.1.11.1

Factor $x^{2} + 25$ out of $x (2 x^{2} - 1) (x^{2} + 25)$ .

$(x^{2} + 25) (x (2 x^{2} - 1)) + (5 x^{2} - 6) (x^{2} + 25) = 0$

Step 2.1.11.2

Factor $x^{2} + 25$ out of $(5 x^{2} - 6) (x^{2} + 25)$ .

$(x^{2} + 25) (x (2 x^{2} - 1)) + (x^{2} + 25) (5 x^{2} - 6) = 0$

Step 2.1.11.3

Factor $x^{2} + 25$ out of $(x^{2} + 25) (x (2 x^{2} - 1)) + (x^{2} + 25) (5 x^{2} - 6)$ .

$(x^{2} + 25) (x (2 x^{2} - 1) + 5 x^{2} - 6) = 0$

Step 2.1.12

Apply the distributive property.

$(x^{2} + 25) (x (2 x^{2}) + x \cdot - 1 + 5 x^{2} - 6) = 0$

Step 2.1.13

Rewrite using the commutative property of multiplication.

$(x^{2} + 25) (2 x \cdot x^{2} + x \cdot - 1 + 5 x^{2} - 6) = 0$

Step 2.1.14

Move $- 1$ to the left of $x$ .

$(x^{2} + 25) (2 x \cdot x^{2} - 1 \cdot x + 5 x^{2} - 6) = 0$

Step 2.1.15

Simplify each term.

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Step 2.1.15.1

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

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Step 2.1.15.1.1

Move $x^{2}$ .

$(x^{2} + 25) (2 (x^{2} x) - 1 \cdot x + 5 x^{2} - 6) = 0$

Step 2.1.15.1.2

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

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Step 2.1.15.1.2.1

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

$(x^{2} + 25) (2 (x^{2} x) - 1 \cdot x + 5 x^{2} - 6) = 0$

Step 2.1.15.1.2.2

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

$(x^{2} + 25) (2 x^{2 + 1} - 1 \cdot x + 5 x^{2} - 6) = 0$

Step 2.1.15.1.3

Add $2$ and $1$ .

$(x^{2} + 25) (2 x^{3} - 1 \cdot x + 5 x^{2} - 6) = 0$

$(x^{2} + 25) (2 x^{3} - 1 x + 5 x^{2} - 6) = 0$

Step 2.1.15.2

Rewrite $- 1 x$ as $- x$ .

$(x^{2} + 25) (2 x^{3} - x + 5 x^{2} - 6) = 0$

Step 2.1.16

Reorder terms.

$(x^{2} + 25) (2 x^{3} + 5 x^{2} - x - 6) = 0$

Step 2.1.17

Factor.

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Step 2.1.17.1

Rewrite $2 x^{3} + 5 x^{2} - x - 6$ in a factored form.

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Step 2.1.17.1.1

Factor $2 x^{3} + 5 x^{2} - x - 6$ using the rational roots test.

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Step 2.1.17.1.1.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1, \pm 2$

Step 2.1.17.1.1.2

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

$\pm 1, \pm 0.5, \pm 6, \pm 3, \pm 2, \pm 1.5$

Step 2.1.17.1.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.1.17.1.1.3.1

Substitute $1$ into the polynomial.

$2 \cdot 1^{3} + 5 \cdot 1^{2} - 1 - 6$

Step 2.1.17.1.1.3.2

Raise $1$ to the power of $3$ .

$2 \cdot 1 + 5 \cdot 1^{2} - 1 - 6$

Step 2.1.17.1.1.3.3

Multiply $2$ by $1$ .

$2 + 5 \cdot 1^{2} - 1 - 6$

Step 2.1.17.1.1.3.4

Raise $1$ to the power of $2$ .

$2 + 5 \cdot 1 - 1 - 6$

Step 2.1.17.1.1.3.5

Multiply $5$ by $1$ .

$2 + 5 - 1 - 6$

Step 2.1.17.1.1.3.6

Add $2$ and $5$ .

$7 - 1 - 6$

Step 2.1.17.1.1.3.7

Subtract $1$ from $7$ .

$6 - 6$

Step 2.1.17.1.1.3.8

Subtract $6$ from $6$ .

$0$

Step 2.1.17.1.1.4

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

$\frac{2 x^{3} + 5 x^{2} - x - 6}{x - 1}$

Step 2.1.17.1.1.5

Divide $2 x^{3} + 5 x^{2} - x - 6$ by $x - 1$ .

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Step 2.1.17.1.1.5.1

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

$x$

$1$

$2 x^{3}$

$5 x^{2}$

$x$

$6$

Step 2.1.17.1.1.5.2

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

					$2 x^{2}$
	$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$

Step 2.1.17.1.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			+	$2 x^{3}$	-	$2 x^{2}$

Step 2.1.17.1.1.5.4

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$

Step 2.1.17.1.1.5.5

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$

Step 2.1.17.1.1.5.6

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$

Step 2.1.17.1.1.5.7

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$7 x$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$

Step 2.1.17.1.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					+	$7 x^{2}$	-	$7 x$

Step 2.1.17.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} - 7 x$

				$2 x^{2}$	+	$7 x$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$

Step 2.1.17.1.1.5.10

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

				$2 x^{2}$	+	$7 x$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$
							+	$6 x$

Step 2.1.17.1.1.5.11

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

				$2 x^{2}$	+	$7 x$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$
							+	$6 x$	-	$6$

Step 2.1.17.1.1.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$7 x$	+	$6$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$
							+	$6 x$	-	$6$

Step 2.1.17.1.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$	+	$6$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$
							+	$6 x$	-	$6$
							+	$6 x$	-	$6$

Step 2.1.17.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 6$

				$2 x^{2}$	+	$7 x$	+	$6$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$
							+	$6 x$	-	$6$
							-	$6 x$	+	$6$

Step 2.1.17.1.1.5.15

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

				$2 x^{2}$	+	$7 x$	+	$6$
$x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$x$	-	$6$
			-	$2 x^{3}$	+	$2 x^{2}$
					+	$7 x^{2}$	-	$x$
					-	$7 x^{2}$	+	$7 x$
							+	$6 x$	-	$6$
							-	$6 x$	+	$6$
										$0$

Step 2.1.17.1.1.5.16

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

$2 x^{2} + 7 x + 6$

Step 2.1.17.1.1.6

Write $2 x^{3} + 5 x^{2} - x - 6$ as a set of factors.

$(x^{2} + 25) ((x - 1) (2 x^{2} + 7 x + 6)) = 0$

Step 2.1.17.1.2

Factor by grouping.

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Step 2.1.17.1.2.1

Factor by grouping.

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Step 2.1.17.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 6 = 12$ and whose sum is $b = 7$ .

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Step 2.1.17.1.2.1.1.1

Factor $7$ out of $7 x$ .

$(x^{2} + 25) ((x - 1) (2 x^{2} + 7 (x) + 6)) = 0$

Step 2.1.17.1.2.1.1.2

Rewrite $7$ as $3$ plus $4$

$(x^{2} + 25) ((x - 1) (2 x^{2} + (3 + 4) x + 6)) = 0$

Step 2.1.17.1.2.1.1.3

Apply the distributive property.

$(x^{2} + 25) ((x - 1) (2 x^{2} + 3 x + 4 x + 6)) = 0$

Step 2.1.17.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.17.1.2.1.2.1

Group the first two terms and the last two terms.

$(x^{2} + 25) ((x - 1) ((2 x^{2} + 3 x) + 4 x + 6)) = 0$

Step 2.1.17.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x^{2} + 25) ((x - 1) (x (2 x + 3) + 2 (2 x + 3))) = 0$

Step 2.1.17.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$(x^{2} + 25) ((x - 1) ((2 x + 3) (x + 2))) = 0$

Step 2.1.17.1.2.2

Remove unnecessary parentheses.

$(x^{2} + 25) ((x - 1) (2 x + 3) (x + 2)) = 0$

Step 2.1.17.2

Remove unnecessary parentheses.

$(x^{2} + 25) (x - 1) (2 x + 3) (x + 2) = 0$

Step 2.2

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

$x^{2} + 25 = 0$

$x - 1 = 0$

$2 x + 3 = 0$

$x + 2 = 0$

Step 2.3

Set $x^{2} + 25$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 25$ equal to $0$ .

$x^{2} + 25 = 0$

Step 2.3.2

Solve $x^{2} + 25 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $25$ from both sides of the equation.

$x^{2} = - 25$

Step 2.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 25}$

Step 2.3.2.3

Simplify $\pm \sqrt{- 25}$ .

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Step 2.3.2.3.1

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Step 2.3.2.3.2

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

Step 2.3.2.3.3

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

$x = \pm i \cdot \sqrt{25}$

Step 2.3.2.3.4

Rewrite $25$ as $5^{2}$ .

$x = \pm i \cdot \sqrt{5^{2}}$

Step 2.3.2.3.5

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

$x = \pm i \cdot 5$

Step 2.3.2.3.6

Move $5$ to the left of $i$ .

$x = \pm 5 i$

Step 2.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5 i$

Step 2.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5 i$

Step 2.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5 i, - 5 i$

Step 2.4

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

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Step 2.4.1

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

$x - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 2.5.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 2.5.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 2.5.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 2.6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.7

The final solution is all the values that make $(x^{2} + 25) (x - 1) (2 x + 3) (x + 2) = 0$ true.

$x = 5 i, - 5 i, 1, - \frac{3}{2}, - 2$

Step 3