Question

Solve by the quadratic formula. List the solutions, separated by commas.
6x^2-18x+12=0
x =

Answer 1

To solve the quadratic equation \(6x^2 - 18x + 12 = 0\) using the quadratic formula, we first identify the coefficients:

- \(a = 6\)
- \(b = -18\)
- \(c = 12\)

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now, let's calculate the discriminant \(b^2 - 4ac\):

\[
b^2 = (-18)^2 = 324
\]
\[
4ac = 4 \cdot 6 \cdot 12 = 288
\]
\[
b^2 - 4ac = 324 - 288 = 36
\]

Now substitute into the quadratic formula:

\[
x = \frac{-(-18) \pm \sqrt{36}}{2 \cdot 6}
\]
\[
x = \frac{18 \pm 6}{12}
\]

Now we can calculate the two possible values for \(x\):

1. First solution:

\[
x = \frac{18 + 6}{12} = \frac{24}{12} = 2
\]

2. Second solution:

\[
x = \frac{18 - 6}{12} = \frac{12}{12} = 1
\]

Thus, the solutions to the equation \(6x^2 - 18x + 12 = 0\) are:

\[
x = 2, 1
\]

Finally, we can write the solutions:

\[
x = 2, 1
\]