Question

Solve 2x² − 3x + 1 = 0 by completing the square. If there are multiple solutions, simplify the solutions and enter them from least to greatest separated by a comma.

Answer 1

To solve the quadratic equation 2x² − 3x + 1 = 0 by completing the square, divide by the coefficient of x², move the constant to the other side, complete the square by adding the squared half of the x-term's coefficient, rewrite as a squared binomial, and solve for x. The solutions are ½ and 1.

Step-by-step explanation:

To solve the equation 2x² − 3x + 1 = 0 by completing the square, follow these steps:

1. Divide the entire equation by 2 to get x² - \(\frac{3}{2}\)x + \(\frac{1}{2}\) = 0.
2. Move the constant term to the other side: x² - \(\frac{3}{2}\)x = -\(\frac{1}{2}\).
3. Add \(\left(\frac{-3}{4}\right)^2 = \(\frac{9}{16}\) to both sides to complete the square: x² - \(\frac{3}{2}\)x + \(\frac{9}{16}\) = \(\frac{9}{16}\) - \(\frac{1}{2}\).
4. Write the left side as a squared binomial and simplify the right side: (x - \(\frac{3}{4}\))^2 = \(\frac{-1}{16}\).
5. Take the square root of both sides, remembering to include both the positive and negative square roots:
   x - \(\frac{3}{4}\) = \(\pm\sqrt{\(\frac{-1}{16}\)}.
6. Solve for x to find the two solutions: x = \(\frac{3}{4}\) \(\pm\sqrt{\(\frac{-1}{16}\)}.

Therefore, the solutions, simplified and ordered from least to greatest, are \(\frac{1}{2}, 1\).