Video Transcript
Which of the following best
describes the roots of the equation 𝑥 squared plus 17 equals zero? Is it (A) two complex roots, (B)
two distinct real roots, (C) one repeated real root, or (D) one repeated nonreal
root?
In order to answer this question,
we will consider a couple of different methods. Firstly, we’ll just solve the
equation 𝑥 squared plus 17 equals zero. Subtracting 17 from both sides, we
have 𝑥 squared is equal to negative 17. Next, we find the square root of
both sides such that 𝑥 is equal to positive or negative the square root of negative
17. We know that the square root of any
negative number is nonreal, and this means we can rule out options (B) and (C).
Recalling that the square root of
negative one is equal to the imaginary, or complex, number 𝑖, then the square root
of negative 17 is equal to root 17𝑖. 𝑥 is therefore equal to the
positive or negative of this, and as such we have two complex roots to the equation
𝑥 squared plus 17 equals zero. They are positive and negative root
17𝑖.
We mentioned earlier that we would
look at a couple of ways of solving this problem. Since our equation is a quadratic,
we can consider the discriminant. For any quadratic equation written
in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, the discriminant is equal to
𝑏 squared minus four 𝑎𝑐. In this question, the coefficient
of 𝑥 squared, our value of 𝑎, is equal to one. There is no 𝑥-term, so 𝑏 is equal
to zero. And the constant 𝑐 is equal to
17. The discriminant is therefore equal
to zero squared minus four multiplied by one multiplied by 17. And this is equal to negative
68.
Recalling that if the discriminant
is less than zero, we have two complex roots. And since negative 68 is indeed
less than zero, the statement that best describes the roots of the equation 𝑥
squared plus 17 equals zero is two complex roots. It is also worth noting that if the
discriminant is greater than zero, we have two distinct real roots. And if the discriminant is equal to
zero, we have one repeated real root.
