Question

Solve by completing the square.
1) x²+10x + 74 = 0
2) 3x² - 6x + 9 = 0

Answer 1

To solve quadratic equations by completing the square, rearrange each to ax² + bx + c = 0, complete the square, and solve for x. Equations 1) and 2) have solutions including imaginary numbers because they lead to negative values inside the square roots after completing the square.

Step-by-step explanation:

To solve the quadratic equations by completing the square, we rearrange each equation into the form ax² + bx + c = 0, then we complete the square for x.

Solution for 1) x² + 10x + 74 = 0:

1. Move the constant term to the other side: x² + 10x = -74.
2. Divide the coefficient of x by 2, square it, and add to both sides: (10/2)² = 25. So, we add 25 to each side to get x² + 10x + 25 = -74 + 25.
3. Now we have (x + 5)² = -49. Take the square root of both sides: x + 5 = ±√(-49). Since the square root of a negative number results in an imaginary number, we get x + 5 = ±7i.
4. Subtract 5 from both sides to get the solution: x = -5 ± 7i.

Solution for 2) 3x² - 6x + 9 = 0:

1. First, divide the entire equation by 3 to make the coefficient of x² equal to 1: x² - 2x + 3 = 0.
2. Next, complete the square as before: (-2/2)² = 1. Add 1 to both sides yielding x² - 2x + 1 = -2 + 1.
3. We obtain (x - 1)² = -1. Taking the square root of both sides, x - 1 = ±√(-1) or x - 1 = ±i.
4. Finally, add 1 to both sides to find the solutions: x = 1 ± i.

In these examples, the solutions include imaginary numbers because the equations have negative values inside the square roots.