Question

3x^2-5x+2=0 by completing the square

Answer 1

To solve the quadratic equation \(3x^2 - 5x + 2 = 0\) by completing the square, we will follow these steps:

1. **Divide by the coefficient of \(x^2\)**: Start by dividing the entire equation by 3 to simplify it.
\[
x^2 - \frac{5}{3}x + \frac{2}{3} = 0
\]

2. **Rearrange the equation**: Move the constant term to the other side of the equation.
\[
x^2 - \frac{5}{3}x = -\frac{2}{3}
\]

3. **Complete the square**: To complete the square, take half of the coefficient of \(x\) (which is \(-\frac{5}{3}\)), square it, and add it to both sides of the equation.
- Half of \(-\frac{5}{3}\) is \(-\frac{5}{6}\).
- Squaring this gives \(\left(-\frac{5}{6}\right)^2 = \frac{25}{36}\).

Add \(\frac{25}{36}\) to both sides:
\[
x^2 - \frac{5}{3}x + \frac{25}{36} = -\frac{2}{3} + \frac{25}{36}
\]

4. **Simplify the right side**: First, convert \(-\frac{2}{3}\) to an equivalent fraction with a denominator of 36:
\[
-\frac{2}{3} = -\frac{24}{36}
\]
Now we can combine:
\[
-\frac{24}{36} + \frac{25}{36} = \frac{1}{36}
\]

5. **Rewrite the left side as a square**: The left side now becomes:
\[
\left(x - \frac{5}{6}\right)^2 = \frac{1}{36}
\]

6. **Take the square root of both sides**:
\[
x - \frac{5}{6} = \pm \frac{1}{6}
\]

7. **Solve for \(x\)**: This gives us two equations:
\[
x - \frac{5}{6} = \frac{1}{6} \quad \text{and} \quad x - \frac{5}{6} = -\frac{1}{6}
\]

- For the first equation:
\[
x = \frac{1}{6} + \frac{5}{6} = 1
\]

- For the second equation:
\[
x = -\frac{1}{6} + \frac{5}{6} = \frac{4}{6} = \frac{2}{3}
\]

8. **Final solutions**: Therefore, the solutions to the equation \(3x^2 - 5x + 2 = 0\) are:
\[
x = 1 \quad \text{and} \quad x = \frac{2}{3}
\]