Question

Given the quadratic equations x^2 - 254x - 12 and x^2 - 7x + 12x^2 - 6x + 5, which of the following statements is true?

a) Both equations have real roots.
b) Both equations have complex roots.
c) The first equation has real roots, and the second has complex roots.
d) The first equation has complex roots, and the second has real roots.

Answer 1

Both quadratic equations have real roots because the discriminants of both equations are positive. The discriminant is calculated using the formula b² - 4ac, and for both equations given, this value is positive.

Step-by-step explanation:

To determine if the quadratic equations have real or complex roots, we look at the discriminant of the quadratic formula, which is given by b² - 4ac. If the discriminant is positive, the equation has real roots; if it's negative, the equation has complex roots; if it's zero, the equation has exactly one real root. For the equation x² - 254x - 12, the discriminant is 254² - 4(1)(-12) which is positive and therefore the equation has real roots. For the second equation, which is x² - 7x + 12x² - 6x + 5, we first simplify it to 13x² - 13x + 5. The discriminant here is (-13)² - 4(13)(5), which is also positive, indicating that this equation too has real roots.