To solve the quadratic equation \(3x^2 - 12x + 12 = 0\), we can use the quadratic formula, which is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 3\), \(b = -12\), and \(c = 12\). We will calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots and then apply the formula to find the solutions.
The given quadratic equation is \(3x^2 - 12x + 12 = 0\). From this equation, we identify the coefficients as follows:
- \(a = 3\)
- \(b = -12\)
- \(c = 12\)
The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \(a\), \(b\), and \(c\), we get:
\[
\Delta = (-12)^2 - 4 \times 3 \times 12 = 144 - 144 = 0
\]
Since the discriminant \(\Delta = 0\), the quadratic equation has exactly one real root (a repeated root).
Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\]
Since \(\Delta = 0\), the formula simplifies to:
\[
x = \frac{-b}{2a}
\]
Substituting the values of \(b\) and \(a\):
\[
x = \frac{-(-12)}{2 \times 3} = \frac{12}{6} = 2
\]
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