Solve by Factoring (x^2-5)^(3/2)=27

${(x^{2} - 5)}^{\frac{3}{2}} = 27$

Step 1

Subtract $27$ from both sides of the equation.

${(x^{2} - 5)}^{\frac{3}{2}} - 27 = 0$

Step 2

Rewrite ${(x^{2} - 5)}^{\frac{3}{2}}$ as ${({(x^{2} - 5)}^{\frac{1}{2}})}^{3}$ .

${({(x^{2} - 5)}^{\frac{1}{2}})}^{3} - 27 = 0$

Step 3

Rewrite $27$ as $3^{3}$ .

${({(x^{2} - 5)}^{\frac{1}{2}})}^{3} - 3^{3} = 0$

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {(x^{2} - 5)}^{\frac{1}{2}}$ and $b = 3$ .

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

Step 5

Simplify.

Multiply the exponents in ${({(x^{2} - 5)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Move $3$ to the left of ${(x^{2} - 5)}^{\frac{1}{2}}$ .

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

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

$({(x^{2} - 5)}^{\frac{1}{2}} - 3) (x^{2} - 5 + 3 {(x^{2} - 5)}^{\frac{1}{2}} + 9) = 0$

Add $- 5$ and $9$ .

$({(x^{2} - 5)}^{\frac{1}{2}} - 3) (x^{2} + 3 {(x^{2} - 5)}^{\frac{1}{2}} + 4) = 0$

Step 6

Simplify $({(x^{2} - 5)}^{\frac{1}{2}} - 3) (x^{2} + 3 {(x^{2} - 5)}^{\frac{1}{2}} + 4)$ .

Expand $({(x^{2} - 5)}^{\frac{1}{2}} - 3) (x^{2} + 3 {(x^{2} - 5)}^{\frac{1}{2}} + 4)$ by multiplying each term in the first expression by each term in the second expression.

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + {(x^{2} - 5)}^{\frac{1}{2}} (3 {(x^{2} - 5)}^{\frac{1}{2}}) + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 {(x^{2} - 5)}^{\frac{1}{2}} {(x^{2} - 5)}^{\frac{1}{2}} + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Multiply ${(x^{2} - 5)}^{\frac{1}{2}}$ by ${(x^{2} - 5)}^{\frac{1}{2}}$ by adding the exponents.

Move ${(x^{2} - 5)}^{\frac{1}{2}}$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 ({(x^{2} - 5)}^{\frac{1}{2}} {(x^{2} - 5)}^{\frac{1}{2}}) + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

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

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 {(x^{2} - 5)}^{\frac{1}{2} + \frac{1}{2}} + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Combine the numerators over the common denominator.

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 {(x^{2} - 5)}^{\frac{1 + 1}{2}} + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Add $1$ and $1$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 {(x^{2} - 5)}^{\frac{2}{2}} + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Divide $2$ by $2$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 {(x^{2} - 5)}^{1} + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Simplify $3 {(x^{2} - 5)}^{1}$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 (x^{2} - 5) + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Apply the distributive property.

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 x^{2} + 3 \cdot - 5 + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Multiply $3$ by $- 5$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 x^{2} - 15 + {(x^{2} - 5)}^{\frac{1}{2}} \cdot 4 - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Move $4$ to the left of ${(x^{2} - 5)}^{\frac{1}{2}}$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 x^{2} - 15 + 4 \cdot {(x^{2} - 5)}^{\frac{1}{2}} - 3 x^{2} - 3 (3 {(x^{2} - 5)}^{\frac{1}{2}}) - 3 \cdot 4 = 0$

Multiply $3$ by $- 3$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 x^{2} - 15 + 4 {(x^{2} - 5)}^{\frac{1}{2}} - 3 x^{2} - 9 {(x^{2} - 5)}^{\frac{1}{2}} - 3 \cdot 4 = 0$

Multiply $- 3$ by $4$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 x^{2} - 15 + 4 {(x^{2} - 5)}^{\frac{1}{2}} - 3 x^{2} - 9 {(x^{2} - 5)}^{\frac{1}{2}} - 12 = 0$

Simplify by adding terms.

Combine the opposite terms in ${(x^{2} - 5)}^{\frac{1}{2}} x^{2} + 3 x^{2} - 15 + 4 {(x^{2} - 5)}^{\frac{1}{2}} - 3 x^{2} - 9 {(x^{2} - 5)}^{\frac{1}{2}} - 12$ .

Subtract $3 x^{2}$ from $3 x^{2}$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} - 15 + 4 {(x^{2} - 5)}^{\frac{1}{2}} + 0 - 9 {(x^{2} - 5)}^{\frac{1}{2}} - 12 = 0$

Add ${(x^{2} - 5)}^{\frac{1}{2}} x^{2} - 15 + 4 {(x^{2} - 5)}^{\frac{1}{2}}$ and $0$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} - 15 + 4 {(x^{2} - 5)}^{\frac{1}{2}} - 9 {(x^{2} - 5)}^{\frac{1}{2}} - 12 = 0$

Subtract $12$ from $- 15$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} - 27 + 4 {(x^{2} - 5)}^{\frac{1}{2}} - 9 {(x^{2} - 5)}^{\frac{1}{2}} = 0$

Subtract $9 {(x^{2} - 5)}^{\frac{1}{2}}$ from $4 {(x^{2} - 5)}^{\frac{1}{2}}$ .

${(x^{2} - 5)}^{\frac{1}{2}} x^{2} - 27 - 5 {(x^{2} - 5)}^{\frac{1}{2}} = 0$

Reorder factors in ${(x^{2} - 5)}^{\frac{1}{2}} x^{2} - 27 - 5 {(x^{2} - 5)}^{\frac{1}{2}}$ .

$x^{2} {(x^{2} - 5)}^{\frac{1}{2}} - 27 - 5 {(x^{2} - 5)}^{\frac{1}{2}} = 0$

Step 7

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 3.74165738, 3.74165738$

Step 8

Solve by Factoring (x^2-5)^(3/2)=27

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