Which value must be added to the expression zĀ² + 12z to make it a perfect-square trinomial?
Answer (2)
To complete the square for the expression z 2 + 12 z , we need to add a value to make it a perfect-square trinomial.
We compare the given expression to the form z 2 + 2 a z + a 2 .
We find that 2 a = 12 , so a = 6 .
The value to be added is a 2 = 6 2 = 36 .
Thus, the value that must be added to the expression is 36 ā .
Explanation
Understanding the Problem We are given the expression z 2 + 12 z and we want to find the value that must be added to it to make it a perfect-square trinomial. A perfect-square trinomial is a trinomial that can be factored into the square of a binomial.
Relating to Perfect-Square Trinomial Form A perfect-square trinomial has the form ( z + a ) 2 = z 2 + 2 a z + a 2 . We want to find a value a 2 such that z 2 + 12 z + a 2 is a perfect-square trinomial.
Finding the Value of 'a' Comparing the given expression z 2 + 12 z with z 2 + 2 a z , we can see that 2 a = 12 . Dividing both sides by 2, we get a = 6 .
Calculating a^2 Now, we need to find a 2 , which is the value that must be added to the expression to make it a perfect-square trinomial. Since a = 6 , we have a 2 = 6 2 = 36 . Therefore, the value that must be added to the expression z 2 + 12 z to make it a perfect-square trinomial is 36.
Verification To verify, we can write the perfect-square trinomial as z 2 + 12 z + 36 . This can be factored as ( z + 6 ) 2 , which is indeed a perfect square.
Examples
Completing the square is a useful technique in algebra. For example, if you're designing a garden and want a square planting area but have some extra space, completing the square helps determine how much additional area you need to create a perfect square. Suppose you have a rectangular area that is z feet wide and (z + 12) feet long. To make it a perfect square, you need to add a certain area to it. By completing the square for the expression z^2 + 12z, you find that adding 36 square feet will make it a perfect square with sides of (z + 6) feet.
To make the expression z 2 + 12 z a perfect-square trinomial, we need to add 36 . This results in the expression becoming z 2 + 12 z + 36 , which can be factored as ( z + 6 ) 2 . Thus, the required value to add is 36 .
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