Question Video: Using the Sign of the Discriminant to Determine the Number of Complex Roots of a Quadratic Equation Mathematics • First Year of Secondary School

Video Transcript

Which is the correct condition for the quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero with real coefficients to have no nonreal roots? (A) The discriminant 𝑏 squared minus four 𝑎𝑐 is positive. (B) The discriminant 𝑏 squared minus four 𝑎𝑐 is nonnegative. Is it (C) the discriminant 𝑏 squared minus four 𝑎𝑐 is equal to zero? (D) The discriminant 𝑏 squared minus four 𝑎𝑐 is negative. Or (E) the discriminant 𝑏 squared minus four 𝑎𝑐 is an integer.

In order to establish which is the correct condition, let’s identify where the discriminant 𝑏 squared minus four 𝑎𝑐 comes from. We have a quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. We’re told the coefficients are real. In other words, 𝑎, 𝑏, and 𝑐 are real numbers.

One of the methods we have to solve an equation of this form is to use the quadratic formula. This quadratic formula tells us that the solution’s are given by the equation 𝑥 equals negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎. Now since this is a quadratic equation, we know 𝑎 can never be equal to zero. If 𝑎 was zero, we’d have a linear equation. And we wouldn’t need the quadratic formula to solve it. So we’re never going to be dividing by 𝑎.

But we do have to be really careful with this expression inside the square root. If this expression is positive, if 𝑏 squared minus four 𝑎𝑐 is greater than zero, then we can evaluate the square root of that expression. This means the square root of 𝑏 squared minus four 𝑎𝑐 will itself be a real number. And so we’ll have two real roots; we’ll have two real values for 𝑥.

If, however, 𝑏 squared minus four 𝑎𝑐, the discriminant, is equal to zero, then we have the square root of zero, which is zero. This means when we substitute everything we know about our quadratic equation into the quadratic formula, we simply get negative 𝑏 over two 𝑎 as the value for 𝑥. So we get one real root.

If, however, the discriminant is less than zero, if it’s negative, then when we find the square root of 𝑏 squared minus four 𝑎𝑐, we are getting a nonreal answer. In other words, we’re going to have an imaginary number. This then, in turn, means that the values for 𝑥 are going to be complex solutions. So we can say that when 𝑏 squared minus four 𝑎𝑐 is less than zero, when it’s negative, we have no real roots.

Now, the question sort of have given us a double negative here, which is the correct condition for the quadratic equation to have no nonreal roots. Another way of thinking about this is, which is the correct condition for the equation to have only real roots? Well, for it to have only real roots, we can say that either the discriminant is positive, in which case it has two real roots, or the discriminant is equal to zero. So it has one real root. In other words, the discriminant that we can use the Δ symbol for must be greater than or equal to zero. Another way to say this is nonnegative. This is a way of explaining that it can be both positive or equal to zero.

And so, as a direct result of the quadratic formula, the discriminant 𝑏 squared minus four 𝑎𝑐 must be nonnegative for our quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero to have no nonreal roots.