To solve the equation [tex]\( 735 = \left(x^2 + x + 111\right)^{\frac{6}{5}} + 6 \)[/tex]:
Step 1: Isolate the term involving [tex]\(x\)[/tex].
[tex]\[
735 - 6 = \left(x^2 + x + 111\right)^{\frac{6}{5}}
\][/tex]
Simplifying the left-hand side:
[tex]\[
729 = \left(x^2 + x + 111\right)^{\frac{6}{5}}
\][/tex]
Step 2: Remove the exponent by raising both sides to the reciprocal power of [tex]\(\frac{6}{5}\)[/tex]:
[tex]\[
729^{\frac{5}{6}} = x^2 + x + 111
\][/tex]
Step 3: Calculate [tex]\(729^{\frac{5}{6}}\)[/tex].
[tex]\[
729 = 3^6 \quad \text{so} \quad 729^{\frac{5}{6}} = (3^6)^{\frac{5}{6}} = 3^5 = 243
\][/tex]
Thus,
[tex]\[
243 = x^2 + x + 111
\][/tex]
Step 4: Form a quadratic equation:
[tex]\[
x^2 + x + 111 = 243
\][/tex]
Subtract 243 from both sides:
[tex]\[
x^2 + x + 111 - 243 = 0
\][/tex]
Simplifying:
[tex]\[
x^2 + x - 132 = 0
\][/tex]
Step 5: Solve the quadratic equation [tex]\(x^2 + x - 132 = 0\)[/tex].
We use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -132\)[/tex].
[tex]\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-132)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{1 + 528}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{529}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm 23}{2}
\][/tex]
This results in two solutions:
1. [tex]\(x = \frac{-1 + 23}{2} = \frac{22}{2} = 11\)[/tex]
2. [tex]\(x = \frac{-1 - 23}{2} = \frac{-24}{2} = -12\)[/tex]
Thus, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
\boxed{11}
\][/tex]
[tex]\[
\boxed{-12}
\][/tex]
