14.Roots of Quadratic Equation

Question 1

What are the roots of a quadratic equation of the form ax^2 + bx + c = 0?

Answer 1

The roots of a quadratic equation can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Question 2

How do you determine the nature of the roots based on the discriminant?

Answer 2

The discriminant (b^2 - 4ac) of the quadratic equation helps determine the nature of the roots. If the discriminant is greater than 0, the equation has two distinct real roots. If the discriminant is equal to 0, the equation has two identical real roots (a perfect square). If the discriminant is less than 0, the equation has two complex (non-real) roots.

Question 3

What is the significance of the ± symbol in the quadratic formula?

Answer 3

The ± symbol indicates that there are two possible solutions for x, one with the positive sign and one with the negative sign. This accounts for the fact that a quadratic equation can have two roots.

Question 4

How do you find the roots if the quadratic equation has complex (non-real) roots?

Answer 4

If the discriminant is negative, the roots of the quadratic equation are complex conjugates. The real part of the roots is -b / (2a), and the imaginary part is ± √( |b^2 - 4ac| ) / (2a)i.

Question 5

What is the sum and product of the roots of a quadratic equation?

Answer 5

For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b / a, and the product of the roots is c / a.

Question 6

Can the quadratic equation have only one real root or no real roots?

Answer 6

Yes, it is possible for a quadratic equation to have one real root (when the discriminant is 0) or no real roots (when the discriminant is negative).