Question

Solve the equation and check it with the theorem of Vieta:

x^2 - 5√2x + 12 = 0
A) x = 2√2, 3√2
B) x = 5√2, -2√2
C) x = -2√2, -3√2
D) x = 3√2, 2√2

Answer 1

The quadratic equation x² - 5√2x + 12 = 0 can be solved using the quadratic formula, yielding the solutions x = 3√2 and x = 2√2, which satisfy Vieta's theorem. The correct option is D.

Step-by-step explanation:

To solve the equation x² - 5√2x + 12 = 0, we can apply the quadratic formula, which is -b ± √b² - 4ac over 2a. Here, a = 1, b = -5√2, and c = 12.

Let us use the quadratic formula:

x = (-(-5√2) ± √(-5√2)² - 4(1)(12)) / (2(1))
= (5√2 ± √(50 - 48)) / 2
= (5√2 ± √2) / 2

The two solutions for x are therefore x = 3√2 and x = 2√2.

To check these solutions with Vietas' theorem, we should have the sum of the roots = -b/a (here it should be 5√2) and the product of the roots = c/a (here it should be 12). For our solutions,
Sum of the roots = 3√2 + 2√2 = 5√2
Product of the roots = 3√2 × 2√2 = 6∙ 2 = 12
This confirms that the solutions are correct and Vieta's theorem is satisfied. The correct answer option is D).