Sure, let's solve the given equation step-by-step:
Given equation:
[tex]\[ 12 - (7x^2 - 14x)^{\frac{5}{5}} = 12 \][/tex]
Step 1: Simplify the exponent [tex]\(\frac{5}{5}\)[/tex].
Since [tex]\(\frac{5}{5}\)[/tex] is equal to 1, we can simplify the equation to:
[tex]\[ 12 - (7x^2 - 14x) = 12 \][/tex]
Step 2: Isolate the term with [tex]\(x\)[/tex].
Subtract 12 from both sides to isolate the term inside the parentheses:
[tex]\[ - (7x^2 - 14x) = 0 \][/tex]
Step 3: Remove the negative sign.
Multiplying both sides by -1 to get:
[tex]\[ 7x^2 - 14x = 0 \][/tex]
Step 4: Factor the quadratic expression.
Factor out the common term (which is [tex]\(7x\)[/tex]) from the left-hand side:
[tex]\[ 7x(x - 2) = 0 \][/tex]
Step 5: Solve for [tex]\(x\)[/tex].
Set each factor equal to zero:
[tex]\[ 7x = 0 \quad \text{or} \quad x - 2 = 0 \][/tex]
From [tex]\(7x = 0\)[/tex]:
[tex]\[ x = 0 \][/tex]
From [tex]\(x - 2 = 0\)[/tex]:
[tex]\[ x = 2 \][/tex]
So, the solutions to the equation [tex]\(12 - (7x^2 - 14x)^{\frac{5}{5}} = 12\)[/tex] are:
[tex]\[ \boxed{0} \][/tex]
[tex]\[ \boxed{2} \][/tex]
Thus,
1. [tex]\(x = 0\)[/tex]
2. [tex]\(x = 2\)[/tex]
These are the solutions to the given equation.
