Final answer:
To make the expression x² + x a perfect-square trinomial, we need to add the square of half of the coefficient of the middle term.
Step-by-step explanation:
To make the expression x² + x a perfect-square trinomial, we need to determine what value should be added to complete the square. In a perfect-square trinomial, the middle term should be equal to two times the product of the square root of the first term and the square root of the last term.
In this case, we have x² + x. The square root of x² is x, and the square root of x is √x. Therefore, the product of these two terms is x * √x, which can be simplified to √x³.
To make the expression a perfect-square trinomial, we need to add the square of half of the coefficient of the middle term, which is (√x³/2)² or (x√x/2)². Therefore, the value that must be added to the expression x² + x to make it a perfect-square trinomial is (x√x/2)².
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