Question

Which of the following statements are true about the quadratic equation x² - 65x + 800 = 0?

A) The quadratic equation has two real and distinct roots.
B) The quadratic equation has two real and equal roots.
C) The quadratic equation has two complex roots.
D) The quadratic equation has no real roots.

Answer 1

The quadratic equation x² - 65x + 800 = 0 has two real and distinct roots.

Statement A) The quadratic equation has two real and distinct roots is true.

Step-by-step explanation:

The quadratic equation x² - 65x + 800 = 0 can be solved by using the quadratic formula.

This formula states that the roots of a quadratic equation ax² + bx + c = 0 are given by:

x = (-b ± √(b² - 4ac)) / (2a)

1. By comparing the given equation to the standard form ax² + bx + c = 0, we can determine that a = 1, b = -65, and c = 800.
2. Using these values, we substitute them into the quadratic formula to find the roots of the equation.
3. Calculating the discriminant, which is the value inside the square root in the formula, will give us information about the type of roots.
4. In this case, the discriminant is equal to (-65)² - 4(1)(800) = 4225 - 3200 = 1025.
5. Since the discriminant is positive, we can conclude that the quadratic equation has two real and distinct roots.
6. Therefore, statement A) The quadratic equation has two real and distinct roots is true.
7. Statements B), C), and D) are false because the equation does not have two real and equal roots, complex roots, or no real roots, respectively.

Answer 2

The quadratic equation x² - 65x + 800 = 0 has two complex roots.

Step-by-step explanation:

The quadratic equation x² - 65x + 800 = 0 can be solved using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).

In this equation, a = 1, b = -65, and c = 800.

By substituting these values into the quadratic formula and simplifying, we find that the discriminant (b² - 4ac) is 65² - 4(1)(800), which is equal to -15935. Since the discriminant is negative, the quadratic equation has two complex roots.