Consider the equation 25𝑥 squared minus 64 equals 𝑎𝑥 plus 𝑏 times 𝑎𝑥 minus 𝑏, where 𝑎 and 𝑏 are constants. Which of the following could be the value of 𝑎? A) eight, B) 64, C) 25, or D) five.
If 25𝑥 squared minus 64 equals 𝑎𝑥 plus 𝑏 times 𝑎𝑥 minus 𝑏, we can use a square method to try and solve 𝑎𝑥 plus 𝑏 times 𝑎𝑥 minus 𝑏. First, we need to multiply 𝑎𝑥 by 𝑎𝑥. This gives us 𝑎 squared 𝑥 squared. Then, we multiply 𝑎𝑥 times 𝑏. And, we get positive 𝑎𝑏𝑥. If we multiply negative 𝑏 times 𝑎𝑥, we get negative 𝑎𝑏𝑥. And, when we multiply negative 𝑏 times positive 𝑏, we get negative 𝑏 squared. We then need to add the four terms in the square, but positive 𝑎𝑏𝑥 plus negative 𝑎𝑏𝑥 cancels out. And so, 𝑎𝑥 plus 𝑏 times 𝑎𝑥 minus 𝑏 equals 𝑎 squared 𝑥 squared minus 𝑏 squared.
25𝑥 squared minus 64 equals 𝑎 squared 𝑥 squared minus 𝑏 squared. We’re interested in solving for 𝑎. But right now, we can only say that 𝑎 squared equals 25. If 𝑎 squared equals 25, we can take the square root of both sides, and then we’ll see that 𝑎 has two results. It is either positive or negative five. Out of our four answer choices, we’re only given positive five. And so, we can say that 𝑎 could be equal to five.
Notice that option C is pretty close. Option C is the value of 𝑎 squared. We also see that 64 as the value of 𝑏 squared. And, if we take the square root of 𝑏 squared and the square root of 64, we’ll see that 𝑏 can be positive or negative eight. So, A is a possible answer for 𝑏. 64 is equal to 𝑏 squared. 25 equals 𝑎 squared. And, option D, five equals 𝑎, which is what we were looking for.