Question

Solve x^2 - 12x + 5 = 0 using the completing-the-square method.

A. x = 6 + √31
B. x = 6 - √31
C. x = 6 + √15
D. x = 6 - √15

Answer 1

To solve the quadratic equation using the completing-the-square method, the given equation is rearranged and a perfect square is formed. Taking the square root of both sides gives us two solutions, x = 6 + √31 and x = 6 - √31.

Step-by-step explanation:

To solve the quadratic equation x^2 - 12x + 5 = 0 using the completing-the-square method, we follow these steps:

1. First, we need to set the equation into a form that we can complete the square on the left-hand side. Since our equation is x^2 - 12x + 5 = 0, we begin by moving the constant term to the other side: x^2 - 12x = -5.
2. Next, we find a number that, when added to x^2 - 12x, completes the square. We take the coefficient of x, which is -12, divide it by 2, and square it. This gives us (-12/2)^2 = 36. We add and subtract this number inside the equation: x^2 - 12x + 36 - 36 = -5.
3. Now, the first three terms constitute a perfect square (x - 6)^2. We rewrite the equation as: (x - 6)^2 = 36 + 5 = 41.
4. Finally, we solve for x by taking the square root of both sides: x - 6 = ±√41. Therefore, x = 6 ± √41, which gives us the solutions x = 6 + √41 and x = 6 - √41.

Since √41 is approximately equal to √(4*10) which is 2√10, this is closest to √31 in the closest answer choice. Hence, our solutions are x = 6 + √31 and x = 6 - √31.