Precalculus Examples

$\sqrt{x^{4} - 2 x^{2} + 1} = 10 x - 22$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{4} - 2 x^{2} + 1}}^{2} = {(10 x - 22)}^{2}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} - 2 x^{2} + 1}$ as ${(x^{4} - 2 x^{2} + 1)}^{\frac{1}{2}}$ .

${({(x^{4} - 2 x^{2} + 1)}^{\frac{1}{2}})}^{2} = {(10 x - 22)}^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(x^{4} - 2 x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(x^{4} - 2 x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{4} - 2 x^{2} + 1)}^{\frac{1}{2} \cdot 2} = {(10 x - 22)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

${(x^{4} - 2 x^{2} + 1)}^{\frac{1}{2} \cdot 2} = {(10 x - 22)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{4} - 2 x^{2} + 1)}^{1} = {(10 x - 22)}^{2}$

Step 2.2.1.2

Simplify.

$x^{4} - 2 x^{2} + 1 = {(10 x - 22)}^{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify ${(10 x - 22)}^{2}$ .

Tap for more steps...

Step 2.3.1.1

Rewrite ${(10 x - 22)}^{2}$ as $(10 x - 22) (10 x - 22)$ .

$x^{4} - 2 x^{2} + 1 = (10 x - 22) (10 x - 22)$

Step 2.3.1.2

Expand $(10 x - 22) (10 x - 22)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.2.1

Apply the distributive property.

$x^{4} - 2 x^{2} + 1 = 10 x (10 x - 22) - 22 (10 x - 22)$

Step 2.3.1.2.2

Apply the distributive property.

$x^{4} - 2 x^{2} + 1 = 10 x (10 x) + 10 x \cdot - 22 - 22 (10 x - 22)$

Step 2.3.1.2.3

Apply the distributive property.

$x^{4} - 2 x^{2} + 1 = 10 x (10 x) + 10 x \cdot - 22 - 22 (10 x) - 22 \cdot - 22$

Step 2.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.1.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$x^{4} - 2 x^{2} + 1 = 10 \cdot 10 x \cdot x + 10 x \cdot - 22 - 22 (10 x) - 22 \cdot - 22$

Step 2.3.1.3.1.2

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

Tap for more steps...

Step 2.3.1.3.1.2.1

Move $x$ .

$x^{4} - 2 x^{2} + 1 = 10 \cdot 10 (x \cdot x) + 10 x \cdot - 22 - 22 (10 x) - 22 \cdot - 22$

Step 2.3.1.3.1.2.2

Multiply $x$ by $x$ .

$x^{4} - 2 x^{2} + 1 = 10 \cdot 10 x^{2} + 10 x \cdot - 22 - 22 (10 x) - 22 \cdot - 22$

Step 2.3.1.3.1.3

Multiply $10$ by $10$ .

$x^{4} - 2 x^{2} + 1 = 100 x^{2} + 10 x \cdot - 22 - 22 (10 x) - 22 \cdot - 22$

Step 2.3.1.3.1.4

Multiply $- 22$ by $10$ .

$x^{4} - 2 x^{2} + 1 = 100 x^{2} - 220 x - 22 (10 x) - 22 \cdot - 22$

Step 2.3.1.3.1.5

Multiply $10$ by $- 22$ .

$x^{4} - 2 x^{2} + 1 = 100 x^{2} - 220 x - 220 x - 22 \cdot - 22$

Step 2.3.1.3.1.6

Multiply $- 22$ by $- 22$ .

$x^{4} - 2 x^{2} + 1 = 100 x^{2} - 220 x - 220 x + 484$

Step 2.3.1.3.2

Subtract $220 x$ from $- 220 x$ .

$x^{4} - 2 x^{2} + 1 = 100 x^{2} - 440 x + 484$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.1.1

Subtract $100 x^{2}$ from both sides of the equation.

$x^{4} - 2 x^{2} + 1 - 100 x^{2} = - 440 x + 484$

Step 3.1.2

Add $440 x$ to both sides of the equation.

$x^{4} - 2 x^{2} + 1 - 100 x^{2} + 440 x = 484$

Step 3.1.3

Subtract $100 x^{2}$ from $- 2 x^{2}$ .

$x^{4} - 102 x^{2} + 1 + 440 x = 484$

Step 3.2

Subtract $484$ from both sides of the equation.

$x^{4} - 102 x^{2} + 1 + 440 x - 484 = 0$

Step 3.3

Subtract $484$ from $1$ .

$x^{4} - 102 x^{2} + 440 x - 483 = 0$

Step 3.4

Factor the left side of the equation.

Tap for more steps...

Step 3.4.1

Factor $x^{4} - 102 x^{2} + 440 x - 483$ using the rational roots test.

Tap for more steps...

Step 3.4.1.1

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.

$p = \pm 1, \pm 483, \pm 3, \pm 161, \pm 7, \pm 69, \pm 21, \pm 23$

$q = \pm 1$

Step 3.4.1.2

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

$\pm 1, \pm 483, \pm 3, \pm 161, \pm 7, \pm 69, \pm 21, \pm 23$

Step 3.4.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

Tap for more steps...

Step 3.4.1.3.1

Substitute $3$ into the polynomial.

$3^{4} - 102 \cdot 3^{2} + 440 \cdot 3 - 483$

Step 3.4.1.3.2

Raise $3$ to the power of $4$ .

$81 - 102 \cdot 3^{2} + 440 \cdot 3 - 483$

Step 3.4.1.3.3

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

$81 - 102 \cdot 9 + 440 \cdot 3 - 483$

Step 3.4.1.3.4

Multiply $- 102$ by $9$ .

$81 - 918 + 440 \cdot 3 - 483$

Step 3.4.1.3.5

Subtract $918$ from $81$ .

$- 837 + 440 \cdot 3 - 483$

Step 3.4.1.3.6

Multiply $440$ by $3$ .

$- 837 + 1320 - 483$

Step 3.4.1.3.7

Add $- 837$ and $1320$ .

$483 - 483$

Step 3.4.1.3.8

Subtract $483$ from $483$ .

$0$

Step 3.4.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 102 x^{2} + 440 x - 483}{x - 3}$

Step 3.4.1.5

Divide $x^{4} - 102 x^{2} + 440 x - 483$ by $x - 3$ .

Tap for more steps...

Step 3.4.1.5.1

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

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

Step 3.4.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

Step 3.4.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

Step 3.4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 3 x^{3}$

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

Step 3.4.1.5.5

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

Step 3.4.1.5.6

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

Step 3.4.1.5.7

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

Step 3.4.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 3.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 9 x^{2}$

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 3.4.1.5.10

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

Step 3.4.1.5.11

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

Step 3.4.1.5.12

Divide the highest order term in the dividend $- 93 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$93 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

Step 3.4.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$93 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

Step 3.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 93 x^{2} + 279 x$

$x^{3}$

$3 x^{2}$

$93 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

Step 3.4.1.5.15

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

$x^{3}$

$3 x^{2}$

$93 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

$161 x$

Step 3.4.1.5.16

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

$x^{3}$

$3 x^{2}$

$93 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

$161 x$

$483$

Step 3.4.1.5.17

Divide the highest order term in the dividend $161 x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$93 x$

$161$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

$161 x$

$483$

Step 3.4.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$93 x$

$161$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

$161 x$

$483$

$161 x$

$483$

Step 3.4.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $161 x - 483$

$x^{3}$

$3 x^{2}$

$93 x$

$161$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

$161 x$

$483$

$161 x$

$483$

Step 3.4.1.5.20

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

$x^{3}$

$3 x^{2}$

$93 x$

$161$

$x$

$3$

$x^{4}$

$0 x^{3}$

$102 x^{2}$

$440 x$

$483$

$x^{4}$

$3 x^{3}$

$102 x^{2}$

$3 x^{3}$

$9 x^{2}$

$93 x^{2}$

$440 x$

$93 x^{2}$

$279 x$

$161 x$

$483$

$161 x$

$483$

$0$

Step 3.4.1.5.21

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

$x^{3} + 3 x^{2} - 93 x + 161$

Step 3.4.1.6

Write $x^{4} - 102 x^{2} + 440 x - 483$ as a set of factors.

$(x - 3) (x^{3} + 3 x^{2} - 93 x + 161) = 0$

Step 3.4.2

Factor $x^{3} + 3 x^{2} - 93 x + 161$ using the rational roots test.

Tap for more steps...

Step 3.4.2.1

Factor $x^{3} + 3 x^{2} - 93 x + 161$ using the rational roots test.

Tap for more steps...

Step 3.4.2.1.1

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.

$p = \pm 1, \pm 161, \pm 7, \pm 23$

$q = \pm 1$

Step 3.4.2.1.2

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

$\pm 1, \pm 161, \pm 7, \pm 23$

Step 3.4.2.1.3

Substitute $7$ and simplify the expression. In this case, the expression is equal to $0$ so $7$ is a root of the polynomial.

Tap for more steps...

Step 3.4.2.1.3.1

Substitute $7$ into the polynomial.

$7^{3} + 3 \cdot 7^{2} - 93 \cdot 7 + 161$

Step 3.4.2.1.3.2

Raise $7$ to the power of $3$ .

$343 + 3 \cdot 7^{2} - 93 \cdot 7 + 161$

Step 3.4.2.1.3.3

Raise $7$ to the power of $2$ .

$343 + 3 \cdot 49 - 93 \cdot 7 + 161$

Step 3.4.2.1.3.4

Multiply $3$ by $49$ .

$343 + 147 - 93 \cdot 7 + 161$

Step 3.4.2.1.3.5

Add $343$ and $147$ .

$490 - 93 \cdot 7 + 161$

Step 3.4.2.1.3.6

Multiply $- 93$ by $7$ .

$490 - 651 + 161$

Step 3.4.2.1.3.7

Subtract $651$ from $490$ .

$- 161 + 161$

Step 3.4.2.1.3.8

Add $- 161$ and $161$ .

$0$

Step 3.4.2.1.4

Since $7$ is a known root, divide the polynomial by $x - 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 3 x^{2} - 93 x + 161}{x - 7}$

Step 3.4.2.1.5

Divide $x^{3} + 3 x^{2} - 93 x + 161$ by $x - 7$ .

Tap for more steps...

Step 3.4.2.1.5.1

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

$x$

$7$

$x^{3}$

$3 x^{2}$

$93 x$

$161$

Step 3.4.2.1.5.2

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

					$x^{2}$
	$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$

Step 3.4.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			+	$x^{3}$	-	$7 x^{2}$

Step 3.4.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 7 x^{2}$

				$x^{2}$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$

Step 3.4.2.1.5.5

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

				$x^{2}$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$

Step 3.4.2.1.5.6

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

				$x^{2}$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$

Step 3.4.2.1.5.7

Divide the highest order term in the dividend $10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$10 x$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$

Step 3.4.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$10 x$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					+	$10 x^{2}$	-	$70 x$

Step 3.4.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{2} - 70 x$

				$x^{2}$	+	$10 x$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$

Step 3.4.2.1.5.10

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

				$x^{2}$	+	$10 x$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$
							-	$23 x$

Step 3.4.2.1.5.11

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

				$x^{2}$	+	$10 x$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$
							-	$23 x$	+	$161$

Step 3.4.2.1.5.12

Divide the highest order term in the dividend $- 23 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$10 x$	-	$23$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$
							-	$23 x$	+	$161$

Step 3.4.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$10 x$	-	$23$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$
							-	$23 x$	+	$161$
							-	$23 x$	+	$161$

Step 3.4.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 23 x + 161$

				$x^{2}$	+	$10 x$	-	$23$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$
							-	$23 x$	+	$161$
							+	$23 x$	-	$161$

Step 3.4.2.1.5.15

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

				$x^{2}$	+	$10 x$	-	$23$
$x$	-	$7$		$x^{3}$	+	$3 x^{2}$	-	$93 x$	+	$161$
			-	$x^{3}$	+	$7 x^{2}$
					+	$10 x^{2}$	-	$93 x$
					-	$10 x^{2}$	+	$70 x$
							-	$23 x$	+	$161$
							+	$23 x$	-	$161$
										$0$

Step 3.4.2.1.5.16

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

$x^{2} + 10 x - 23$

Step 3.4.2.1.6

Write $x^{3} + 3 x^{2} - 93 x + 161$ as a set of factors.

$(x - 3) ((x - 7) (x^{2} + 10 x - 23)) = 0$

Step 3.4.2.2

Remove unnecessary parentheses.

$(x - 3) (x - 7) (x^{2} + 10 x - 23) = 0$

Step 3.5

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

$x - 3 = 0$

$x - 7 = 0$

$x^{2} + 10 x - 23 = 0$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$x - 3 = 0$

Step 3.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.7

Set $x - 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.7.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.7.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.8

Set $x^{2} + 10 x - 23$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.8.1

Set $x^{2} + 10 x - 23$ equal to $0$ .

$x^{2} + 10 x - 23 = 0$

Step 3.8.2

Solve $x^{2} + 10 x - 23 = 0$ for $x$ .

Tap for more steps...

Step 3.8.2.1

Use the quadratic formula to find the solutions.

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

Step 3.8.2.2

Substitute the values $a = 1$ , $b = 10$ , and $c = - 23$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot - 23)}}{2 \cdot 1}$

Step 3.8.2.3

Simplify.

Tap for more steps...

Step 3.8.2.3.1

Simplify the numerator.

Tap for more steps...

Step 3.8.2.3.1.1

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

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Step 3.8.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 23$ .

Tap for more steps...

Step 3.8.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot - 23}}{2 \cdot 1}$

Step 3.8.2.3.1.2.2

Multiply $- 4$ by $- 23$ .

$x = \frac{- 10 \pm \sqrt{100 + 92}}{2 \cdot 1}$

Step 3.8.2.3.1.3

Add $100$ and $92$ .

$x = \frac{- 10 \pm \sqrt{192}}{2 \cdot 1}$

Step 3.8.2.3.1.4

Rewrite $192$ as $8^{2} \cdot 3$ .

Tap for more steps...

Step 3.8.2.3.1.4.1

Factor $64$ out of $192$ .

$x = \frac{- 10 \pm \sqrt{64 (3)}}{2 \cdot 1}$

Step 3.8.2.3.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 10 \pm \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 3.8.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 10 \pm 8 \sqrt{3}}{2 \cdot 1}$

Step 3.8.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 8 \sqrt{3}}{2}$

Step 3.8.2.3.3

Simplify $\frac{- 10 \pm 8 \sqrt{3}}{2}$ .

$x = - 5 \pm 4 \sqrt{3}$

Step 3.8.2.4

The final answer is the combination of both solutions.

$x = - 5 + 4 \sqrt{3}, - 5 - 4 \sqrt{3}$

Step 3.9

The final solution is all the values that make $(x - 3) (x - 7) (x^{2} + 10 x - 23) = 0$ true.

$x = 3, 7, - 5 + 4 \sqrt{3}, - 5 - 4 \sqrt{3}$

Step 4

Exclude the solutions that do not make $\sqrt{x^{4} - 2 x^{2} + 1} = 10 x - 22$ true.

$x = 3, 7$