Which quadratic equation has roots 𝑥 is equal to positive or negative three 𝑖? Is it (A) 𝑥 squared is equal to negative three, (B) 𝑥 squared is equal to three, (C) 𝑥 squared is equal to negative six, (D) 𝑥 squared is equal to negative nine, or (E) 𝑥 squared is equal to nine?
One way of answering this question would be to work out the two roots of each of our five quadratics and see which one is equal to positive and negative three 𝑖. We would do this by square rooting both sides of the equations. Alternatively, we can start with the fact that the roots are 𝑥 equals three 𝑖 and 𝑥 equals negative three 𝑖. Subtracting three 𝑖 from both sides of the first equation gives us 𝑥 minus three 𝑖 is equal to zero. This means that 𝑥 minus three 𝑖 is a factor of our quadratic equation. We can add three 𝑖 to both sides of our second equation such that 𝑥 plus three 𝑖 is equal to zero. This means that 𝑥 plus three 𝑖 is also a factor of our quadratic.
Multiplying these two equations, we see that the product of the two factors must equal zero. 𝑥 minus three 𝑖 multiplied by 𝑥 plus three 𝑖 is equal to zero. We can then expand the brackets or distribute the parentheses using the FOIL method. 𝑥 multiplied by 𝑥 is equal to 𝑥 squared. Multiplying the outer terms, we have three 𝑥𝑖. Multiplying the inner terms, we have negative three 𝑥𝑖. Finally, multiplying the last terms gives us negative nine 𝑖 squared. 𝑥 squared plus three 𝑥𝑖 minus three 𝑥𝑖 minus nine 𝑖 squared is equal to zero. The middle two terms cancel.
We also recall that 𝑖 squared is equal to negative one. Our equation simplifies to 𝑥 squared minus nine multiplied by negative one is equal to zero. This in turn gives us 𝑥 squared plus nine equals zero. Subtracting nine from both sides of this equation, we have 𝑥 squared is equal to negative nine. This means that the correct answer is option (D). The quadratic equation 𝑥 squared is equal to negative nine has roots 𝑥 is equal to positive or negative three 𝑖.
This question leads us to a general rule when trying to solve a quadratic that is the sum of two squares. The expression 𝑥 squared plus 𝑎 squared can be factored into two sets of parentheses, 𝑥 plus 𝑎𝑖 and 𝑥 minus 𝑎𝑖. In this question, 𝑥 squared plus nine is the same as 𝑥 plus three 𝑖 multiplied by 𝑥 minus three 𝑖.