Question

Which statement concerning the equation x² - 1 = x is true?

1. Its discriminant is 0, so it has no solution.

2. Its discriminant is 5, so it has two real solutions.

3. Its discriminant is 0, so it has one real solution.

4. Its discriminant is -3, so it has two complex solutions.

Answer 1

The equation x² - 1 = x can be rewritten as x² - x - 1 = 0. We can use the quadratic formula to find the solutions:

x = (-b ± sqrt(b² - 4ac)) / 2a

In this case, a = 1, b = -1, and c = -1. So:

x = (1 ± sqrt(1 - 4(1)(-1))) / 2(1)

x = (1 ± sqrt(5)) / 2

Therefore, the equation has two real solutions, and the discriminant is positive. The answer is 2. Its discriminant is 5, so it has two real solutions.

Answer 2

The statement "Its discriminant is 5, so it has two real solutions" is not true for the equation x² - 1 = x.

The discriminant of a quadratic equation of the form ax² + bx + c = 0 is given by the expression b² - 4ac. In this case, the equation can be rewritten as x² - x - 1 = 0, so a = 1, b = -1, and c = -1. Therefore, the discriminant is (-1)² - 4(1)(-1) = 5.

Since the discriminant is positive, the equation has two real solutions. However, this statement does not apply to the given equation because it is not in standard quadratic form.

To solve the equation x² - 1 = x, we can rearrange it as x² - x - 1 = 0 and then use the quadratic formula to find its solutions. The quadratic formula is x = (-b ± sqrt(b² - 4ac)) / 2a. Plugging in the values a = 1, b = -1, and c = -1, we get x = (1 ± sqrt(5)) / 2. Therefore, the equation has two real solutions, which are x = (1 + sqrt(5)) / 2 and x = (1 - sqrt(5)) / 2.