Solve for x 2(x^2+x-11)^(3/2)=54

$2 {(x^{2} + x - 11)}^{\frac{3}{2}} = 54$

Divide each term in $2 {(x^{2} + x - 11)}^{\frac{3}{2}} = 54$ by $2$ and simplify.

Divide each term in $2 {(x^{2} + x - 11)}^{\frac{3}{2}} = 54$ by $2$ .

$\frac{2 {(x^{2} + x - 11)}^{\frac{3}{2}}}{2} = \frac{54}{2}$

Simplify the left side.

Cancel the common factor.

$\frac{2 {(x^{2} + x - 11)}^{\frac{3}{2}}}{2} = \frac{54}{2}$

Divide ${(x^{2} + x - 11)}^{\frac{3}{2}}$ by $1$ .

${(x^{2} + x - 11)}^{\frac{3}{2}} = \frac{54}{2}$

Simplify the right side.

Divide $54$ by $2$ .

${(x^{2} + x - 11)}^{\frac{3}{2}} = 27$

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

${({(x^{2} + x - 11)}^{\frac{3}{2}})}^{\frac{2}{3}} = 27^{\frac{2}{3}}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x^{2} + x - 11)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Multiply the exponents in ${({(x^{2} + x - 11)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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

${(x^{2} + x - 11)}^{\frac{3}{2} \cdot \frac{2}{3}} = 27^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x^{2} + x - 11)}^{\frac{3}{2} \cdot \frac{2}{3}} = 27^{\frac{2}{3}}$

Rewrite the expression.

${(x^{2} + x - 11)}^{\frac{1}{2} \cdot 2} = 27^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + x - 11)}^{\frac{1}{2} \cdot 2} = 27^{\frac{2}{3}}$

Rewrite the expression.

${(x^{2} + x - 11)}^{1} = 27^{\frac{2}{3}}$

Simplify.

$x^{2} + x - 11 = 27^{\frac{2}{3}}$

Simplify the right side.

Simplify $27^{\frac{2}{3}}$ .

Simplify the expression.

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

$x^{2} + x - 11 = {(3^{3})}^{\frac{2}{3}}$

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

$x^{2} + x - 11 = 3^{3 (\frac{2}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{2} + x - 11 = 3^{3 (\frac{2}{3})}$

Rewrite the expression.

$x^{2} + x - 11 = 3^{2}$

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

$x^{2} + x - 11 = 9$

Solve for $x$ .

Subtract $9$ from both sides of the equation.

$x^{2} + x - 11 - 9 = 0$

Subtract $9$ from $- 11$ .

$x^{2} + x - 20 = 0$

Factor $x^{2} + x - 20$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$(x - 4) (x + 5) = 0$

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

$x - 4 = 0$

$x + 5 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x - 4) (x + 5) = 0$ true.

$x = 4, - 5$

Solve for x 2(x^2+x-11)^(3/2)=54

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