Question

Solve the equation y = x² - 8x + 19 by completing the square.

Answer 1

To solve the equation y = x² - 8x + 19 by completing the square, follow these steps: move the constant term to the other side, add and subtract the squared value of half the coefficient of x, group the perfect square terms, simplify, isolate the perfect square term, take the square root, and solve for x.

Step-by-step explanation:

To solve the equation y = x² - 8x + 19 by completing the square, follow these steps:

1. Move the constant term (19) to the other side of the equation, making it equal to zero: x² - 8x + 19 = 0
2. Take half of the coefficient of x (-8) and square it to get -16. Add and subtract this value inside the parentheses: x² - 8x + (-16) + 16 + 19 = 0
3. Rearrange the equation to group the perfect square terms and simplify: (x² - 8x + 16) + 19 - 16 = 0
4. Factor the perfect square trinomial and combine like terms: (x - 4)² + 19 - 16 = 0
5. Simplify further to isolate the perfect square term: (x - 4)² + 3 = 0
6. Subtract 3 from both sides: (x - 4)² = -3
7. Take the square root of both sides, remembering to include both the positive and negative root: x - 4 = ±√(-3)
8. Add 4 to both sides to solve for x: x = 4 ± √(-3)