Solve the quadratic equation 3x^2 - 5x + 2 = 0 using the quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

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Step-by-Step Solution

Step 1: Identify coefficients

First, we compare the given quadratic equation $3x^2 - 5x + 2 = 0$ with the standard form of a quadratic equation, $ax^2 + bx + c = 0$. By comparing the coefficients, we identify the values for $a$, $b$, and $c$.

Step 2: Substitute into the quadratic formula

Now, we substitute the identified values of $a=3$, $b=-5$, and $c=2$ into the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This sets up the equation for solving $x$.

Step 3: Simplify the expression under the square root

We simplify the expression under the square root, which is the discriminant. First, we calculate $(-5)^2 = 25$ and $4(3)(2) = 24$. Then, we subtract these values to get $25 - 24 = 1$.

Step 4: Calculate the square root

The square root of $1$ is $1$. We replace $\sqrt{1}$ with $1$ in the formula.

Step 5: Find the two possible values for x

Finally, we calculate the two possible values for $x$. For the first value, we use the plus sign: $x_1 = \frac{5 + 1}{6} = \frac{6}{6} = 1$. For the second value, we use the minus sign: $x_2 = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}$.