What constant needs to be added to the expression x² + 6x to create a perfect square trinomial?

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To determine what constant should be added to the expression x² + 6x in order to form a perfect square trinomial, we can follow a systematic approach.

A perfect square trinomial can be expressed in the form (a + b)², which expands to a² + 2ab + b². In our case, we need to rewrite the expression such that it fits this format.

We start with the first two terms of our expression:

x² + 6x

Here, we can observe:

  • a is the coefficient of x², which is simply x.
  • 2ab corresponds to the term 6x. To find b, we can set up the equation:

2ab = 6x

We can simplify this to:

2(x)(b) = 6x

Dividing both sides by 2x gives us:

b = 3

Now, we know that (x + 3)² would produce the desired form, but we must add b² to complete the perfect square trinomial:

b² = 3² = 9

Thus, to form a perfect square trinomial from x² + 6x, we need to add 9.

Finally, our complete perfect square trinomial will be:

x² + 6x + 9 = (x + 3)²

In summary, the constant that should be added to the expression x² + 6x to create a perfect square trinomial is 9.

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