Question

By first completing the square, solve x² - 3x 1/4 = 0. give your answers fully simplified in the form = a±√b, where a a

Answer 1

To solve the quadratic equation by completing the square, we adjust the equation to (x - 3/2)² = -1/36 and take the square root of both sides, remembering to include the negative root, achieving the solution x = 3/2 ± i/6.

Step-by-step explanation:

To solve the quadratic equation x² - 3x + ¼ = 0 by completing the square, we would follow these steps:

1. Rewrite the equation in the form ax² + bx + c = 0.
2. Move the constant term to the other side: x² - 3x = -¼.
3. Find the number that completes the square for the expression on the left side. This is (b/2a)² = (-3/2*1)² = ¹/₉.
4. Add this number to both sides of the equation, resulting in x² - 3x + ¹/₉ = -¼ + ¹/₉.
5. Write the left side as a squared binomial: (x - 3/2)².
6. Simplify the right side by finding a common denominator and combining the fractions: -¼ + ¹/₉ = -1/36.
7. The equation is now (x - 3/2)² = -1/36.
8. Take the square root of both sides, remembering to include both the positive and negative roots: x - 3/2 = ±√(-1/36).
9. Add 3/2 to both sides, resulting in the final simplified answer: x = 3/2 ± √(-1/36).
10. The final step is to calculate the square root of -1/36. Since the square root of a negative number involves an 'i' (the imaginary unit), we get x = 3/2 ± √i/6.

Therefore, the solutions to the original equation are x = 3/2 ± √i/6.