For which values of 𝑘 is 16𝑥 squared plus 𝑘𝑥 plus 81 a perfect square?
This expression is called a trinomial as it has three terms. In order for a trinomial of this form to be a perfect square, it must fulfill two conditions. Firstly, the first and last terms must individually be perfect squares. Secondly, the middle term must be twice the product of their square roots. Let’s see how to use these two conditions to find the value of 𝑘.
Firstly, we’ll consider the first and last terms which must be perfect squares. We know that 81 is a perfect square. It’s equal to nine squared. Remember it’s also equal to negative nine squared. So we could write 81 as plus or minus nine squared.
What about the first term — 16𝑥 squared? This is also a perfect square. It’s equal to four 𝑥 all squared. Four squared is 16 and 𝑥 squared is 𝑥 squared. It could also be written as negative four 𝑥 all squared. So again, we have plus or minus four 𝑥 squared.
So the first condition for this trinomial to be a perfect square is fulfilled. How does this help us with finding the value of 𝑘? Well, remember the middle term must be twice the product of the square roots of the first and last terms. As an equation then, we have that the middle term, which is 𝑘𝑥, is equal to two multiplied by plus or minus the square root of the first and last terms — the square root of 16𝑥 squared multiplied by the square root of 81.
Remember the square roots are four 𝑥 and nine. So bringing the plus or minus to the front of this equation, we have 𝑘𝑥 is equal to plus or minus two multiplied by four 𝑥 multiplied by nine. Simplifying the right-hand side gives 𝑘𝑥 is equal to plus or minus 72𝑥. We can cancel our factor of 𝑥 from both sides of the equation, giving 𝑘 is equal to plus or minus 72.
The values of 𝑘 for which the trinomial 16𝑥 squared plus 𝑘𝑥 plus 81 is a perfect square are 72 and negative 72.