Replace ∗ with a monomial so that the derived expression may be represented as a square of a binomial: 16x^2+24xy + ∗

9y²

Step-by-step explanation:

The square of a binomial can be written as (a + b)².  If you expand this formula, you get

(a + b)(a + b).  Use F.O.I.L. to multiply this out..

F - stands for 'Firsts' (the first values in each set of parenthesis)

O - stands for 'Outsides'  ( the first value in the first set of parenthesis, and the

                                           second value in the second set of parenthesis)

I - stands for 'Insides'  ( the second value in the first set of parenthesis, and

                                       the first value in the second set of parenthesis)

L - stands for 'Lasts'   ( the second value in each set of parenthesis)

This is the order you multiply them in...so we get

F -  a²     (a times a)

O - ab    

I  -  ba    (which we rewrite as ab, since order doesn't matter when multiplying)

L  -  b²

           We add them together to get   a² + ab + ab + b²

which simplifies to a² + 2ab + b²

Read this as, the square of the first term in the parenthesis (a²), plus twice the product of the terms (2ab is twice the product of a and b), plus the square of the last term in the parenthesis)

So to solve this we need to know what makes 16x^2+24xy + ∗ factor into some form of (a + b)².

Look at the bold paragraph above and work backwards.

Take the square root of the term 16x², which is 4x.  That goes in the first value of the parenthesis, so we have

    (4x + c)²    

We know that 24xy is twice the product of the two terms, so we divide by 2 to see the product of the two terms.

  24xy/2  = 12xy,

We know one term, so divide 12xy by the first term, 4x to find the second term.

  12xy/4x = 3y

So the last term in our set of parentesis is 3y, so we have

   (4x + 3y)²  

To get the missing value, square the second term, (3y)² = 9y²

So 9y² is our missing term.

16x^2+24xy + 9y² = (4x + 3y)²

idn

step-by-step explanation: