Video Transcript
If the expression one-third times 𝑥 plus 𝑐 times 𝑥 minus 𝑐, where 𝑐 is a positive constant, can be written as one-third 𝑥 squared minus three, what is the value of 𝑐?
What we’re saying is that one-third 𝑥 plus 𝑐 times 𝑥 minus 𝑐 is equal to one-third 𝑥 squared minus three. In order to solve for 𝑐, we need to expand this value. To do that, you could FOIL, multiplying 𝑥 times 𝑥 to get 𝑥 squared the first, then the outer, 𝑥 times negative 𝑐 equals negative 𝑐𝑥, then multiply the inner two terms, 𝑐 times 𝑥 equals positive 𝑐𝑥, and 𝑐 times negative 𝑐 equals negative 𝑐 squared. We wanna combine our like terms here, 𝑐𝑥 minus 𝑐𝑥 cancels out, and we have 𝑥 squared minus 𝑐 squared.
We can’t forget to bring down that one-third. And, now we wanna distribute this one-third to the 𝑥 squared and to the negative 𝑐 squared. We’ll have one-third 𝑥 squared minus one-third 𝑐 squared equals one-third 𝑥 squared minus three. We have one-third 𝑥 squared on both sides of the equation and that cancels out, leaving us with one-third 𝑐 squared equals negative three.
To get rid of that negative one-third, we can multiply both sides of the equation by negative three. Negative three times negative one-third cancels out. And, we have 𝑐 squared equals negative three times negative three. 𝑐 squared equals nine. And, to find 𝑐, we take the square root of both sides. The square root of 𝑐 squared equals 𝑐, and the square root of nine is plus or minus three. However, we’ve already been told that 𝑐 is a positive constant, and that means 𝑐 equals positive three.
