Video: US-SAT05S3-Q03-676198594981

Consider the equation 25π‘₯Β² βˆ’ 64 = (π‘Žπ‘₯ + 𝑏)(π‘Žπ‘₯ βˆ’ 𝑏), where π‘Ž and 𝑏 are constants. Which of the following could be the value of π‘Ž? [A] 8 [B] 64 [C] 25 [D] 5.

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Video Transcript

Consider the equation 25π‘₯ squared minus 64 equals π‘Žπ‘₯ plus 𝑏 times π‘Žπ‘₯ minus 𝑏, where π‘Ž and 𝑏 are constants. Which of the following could be the value of π‘Ž? A) eight, B) 64, C) 25, or D) five.

If 25π‘₯ squared minus 64 equals π‘Žπ‘₯ plus 𝑏 times π‘Žπ‘₯ minus 𝑏, we can use a square method to try and solve π‘Žπ‘₯ plus 𝑏 times π‘Žπ‘₯ minus 𝑏. First, we need to multiply π‘Žπ‘₯ by π‘Žπ‘₯. This gives us π‘Ž squared π‘₯ squared. Then, we multiply π‘Žπ‘₯ times 𝑏. And, we get positive π‘Žπ‘π‘₯. If we multiply negative 𝑏 times π‘Žπ‘₯, we get negative π‘Žπ‘π‘₯. 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 π‘Žπ‘π‘₯ plus negative π‘Žπ‘π‘₯ cancels out. And so, π‘Žπ‘₯ plus 𝑏 times π‘Žπ‘₯ minus 𝑏 equals π‘Ž squared π‘₯ squared minus 𝑏 squared.

25π‘₯ squared minus 64 equals π‘Ž squared π‘₯ squared minus 𝑏 squared. We’re interested in solving for π‘Ž. But right now, we can only say that π‘Ž squared equals 25. If π‘Ž squared equals 25, we can take the square root of both sides, and then we’ll see that π‘Ž 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 π‘Ž could be equal to five.

Notice that option C is pretty close. Option C is the value of π‘Ž 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 π‘Ž squared. And, option D, five equals π‘Ž, which is what we were looking for.

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