Solve for x (x^2+4x+5)^(3/2)+1=0

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

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

Step 3

Simplify the exponent.

Simplify the left side.

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

Multiply the exponents in ${({(x^{2} + 4 x + 5)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(- 1)}^{\frac{2}{3}}$ .

Simplify the expression.

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{2} + 4 x + 5 = {(- 1)}^{2}$

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

$x^{2} + 4 x + 5 = 1$

Step 4

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} + 4 x + 5 - 1 = 0$

Subtract $1$ from $5$ .

$x^{2} + 4 x + 4 = 0$

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

$x^{2} + 4 x + 2^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 2 + 2^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

${(x + 2)}^{2} = 0$

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Exclude the solutions that do not make ${(x^{2} + 4 x + 5)}^{\frac{3}{2}} + 1 = 0$ true.

$No solution$

Solve for x (x^2+4x+5)^(3/2)+1=0

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