Let's solve the given equation step-by-step.
The given equation is:
[tex]\[
(3x - 2)^2 - 4 = 9x - 6
\][/tex]
First, expand the left side of the equation:
[tex]\[
(3x - 2)^2 - 4 = (3x - 2)(3x - 2) - 4
\][/tex]
[tex]\[
(3x - 2)(3x - 2) = 9x^2 - 6x - 6x + 4 = 9x^2 - 12x + 4
\][/tex]
So, substituting back, we get:
[tex]\[
9x^2 - 12x + 4 - 4 = 9x - 6
\][/tex]
Simplify by combining like terms:
[tex]\[
9x^2 - 12x = 9x - 6
\][/tex]
Move all terms to one side to set the equation to zero:
[tex]\[
9x^2 - 12x - 9x + 6 = 0
\][/tex]
[tex]\[
9x^2 - 21x + 6 = 0
\][/tex]
Now we have a quadratic equation:
[tex]\[
9x^2 - 21x + 6 = 0
\][/tex]
To solve this quadratic equation, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 9 \)[/tex], [tex]\( b = -21 \)[/tex], and [tex]\( c = 6 \)[/tex].
First, calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
[tex]\[
\Delta = (-21)^2 - 4 \cdot 9 \cdot 6
\][/tex]
[tex]\[
\Delta = 441 - 216
\][/tex]
[tex]\[
\Delta = 225
\][/tex]
Next, compute the solutions using the quadratic formula:
[tex]\[
x = \frac{-(-21) \pm \sqrt{225}}{2 \cdot 9}
\][/tex]
[tex]\[
x = \frac{21 \pm 15}{18}
\][/tex]
This gives us two potential solutions:
[tex]\[
x = \frac{21 + 15}{18} = \frac{36}{18} = 2
\][/tex]
[tex]\[
x = \frac{21 - 15}{18} = \frac{6}{18} = \frac{1}{3}
\][/tex]
Therefore, the two solutions to the equation are:
[tex]\[
x = 2 \quad \text{and} \quad x = \frac{1}{3}
\][/tex]
Thus, one value of [tex]\( x \)[/tex] that is a solution to the equation is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
