What number could be placed in the blank to make the following a perfect-square trinomial?
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Answer (2)
We have the expression x 2 + 8 x .
To form a perfect square trinomial, we need to add ( 2 8 ) 2 .
Calculate ( 2 8 ) 2 = 4 2 = 16 .
Since 16 is not in the options, there is likely an error in the question. If 16 was an option, the answer would be 16 .
Explanation
Understanding the Problem We are given the expression x 2 + 8 x and asked to find a constant term that, when added to the expression, makes it a perfect square trinomial. A perfect square trinomial has the form ( x + a ) 2 = x 2 + 2 a x + a 2 . We need to find the value of a 2 that completes the square.
Finding the Value of a Comparing x 2 + 8 x with x 2 + 2 a x , we can see that 2 a = 8 . Solving for a , we get a = 2 8 = 4 .
Calculating a^2 Now, we need to find the value of a 2 . Since a = 4 , we have a 2 = 4 2 = 16 .
Checking the Options We are given the options 0, 4, 25, and 100. We need to check which of these options is equal to 16. None of the given options is equal to 16. However, there seems to be a mistake in the problem. The correct question should be: Which of the following terms can be added to x 2 + 8 x to make a perfect-square trinomial? The options are 0, 4, 16, 25, 100. In this case, the answer is 16. But since 16 is not in the original options, let's re-examine the problem.
We have x 2 + 8 x . To complete the square, we need to add ( 2 8 ) 2 = 4 2 = 16 . Since 16 is not among the options, there might be an error in the question or the options provided. However, if we proceed with the given options, we need to choose the one that is closest to 16. The closest option is 25, but it's not the correct answer. Let's assume there was a typo and 16 was meant to be one of the options. Then the correct answer would be 16.
Conclusion Since 16 is not among the options, and we are forced to choose from the given options, we must conclude there is an error in the problem statement or the options. However, based on our calculations, the correct term to add is 16. Since none of the options match, we cannot provide a correct answer from the given choices. If 16 was an option, we would have ( x + 4 ) 2 = x 2 + 8 x + 16 .
Examples
Perfect square trinomials are useful in many areas of mathematics and engineering. For example, in physics, when analyzing projectile motion, completing the square helps determine the maximum height reached by the projectile. In engineering, it can be used to optimize designs or analyze systems. For instance, if you have a quadratic equation representing the cost of production, completing the square can help you find the production level that minimizes the cost. This technique is also used in computer graphics to create smooth curves and surfaces.
To complete the expression into a perfect-square trinomial, the number that should be placed in the blank is 16 . This is derived from finding the value of a such that x 2 + 8 x matches the form x 2 + 2 a x , leading to a 2 = 16 . Hence, the answer is 16 .
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