x^2+10x

Add the proper constant to the binomial so tha tthe resulting trinomial is a perfect square trinomial. Then factor the trinomial

To turn the given binomial, x^2 + 10x, into a perfect square trinomial, we need to find the constant term that, when added to the binomial, will make it a perfect square.

To do this, we take half of the coefficient of the x term (which is 10 in this case), square it, and add it to the binomial.

Step 1: Take half of the coefficient of the x term: 10/2 = 5
Step 2: Square the result from Step 1: 5^2 = 25

Now, we add the constant term of 25 to the binomial:
x^2 + 10x + 25

The resulting trinomial, x^2 + 10x + 25, is a perfect square trinomial.

To factor the trinomial, we can use the fact that a perfect square trinomial can be written as the square of a binomial.

The square root of the first term, x^2, is x.
The square root of the last term, 25, is 5.

So, the factored form of the trinomial x^2 + 10x + 25 is:
(x + 5)(x + 5), or alternatively, (x + 5)^2.

Example:

x²+6x
=x²+2(3x)
=(x+3)²-3²
Or:
x²+6x+3² = (x+3)²