Question

What does it mean if the discriminant of a quadratic is negative?

Answer 1

Hint:We should just remember the quadratic formula that is used to find the roots of a quadratic problem whose pairs cannot be easily made and roots could be found by that method basically. So, for that we use this discriminant given by: $D=\sqrt{b^2-4ac}$. We need to tell what happens if the discriminant is negative or what kind of roots we get when the discriminant is negative.

Complete step-by-step solution:
Suppose that we are given a polynomial as $a{x}^2+bx+c=0$ so the discriminant of this quadratic polynomial is given by $D=\sqrt{b^2-4ac}$ where we can see that a is the coefficient of $x^2$, b is the coefficient of x and c is the constant term of the taken polynomial.
So, by using this discriminant property we can have three cases when the discriminant is positive, negative or zero.
Now, when discriminant is positive, we say that we have two distinct real roots of the given polynomial.
When the discriminant is zero, we say that we have two equal real roots of the given polynomial and when the discriminant is negative, we say that we will get two complex roots in nature for the given polynomial.
So, using the discriminant we can identify the type of roots we will get and hence easily find the roots of the polynomial using the formula $\text{sol}={\displaystyle \frac{-b\pm D}{2a}}$ directly substituting the values in this formula will give the answer.

Note: While solving the quadratic equations we use this formula mainly when we are not able to find the factors of the polynomials in order to find the roots. So we find using this formula and value of a,b,c must we clearly and carefully substituted.