Question

Replace ∗ with a monomial to obtain an identity: (∗− 2m)^2 = 100–40m+4m^2

and (3a+2.5b)^2=9a^2+6.25b^2 + ∗

Answer 1

To complete the binomial squares, the monomials that make the identities true are 10 for the first equation, resulting in the identity (10−2m)^2, and 15ab for the second equation, rendering the identity (3a+2.5b)^2 with the included middle term 15ab.

Step-by-step explanation:

The question involves finding a missing monomial in two binomial square expressions.

Taking the first expression, ∗−2m)^2 = 100−40m+4m^2, we know that the binomial square ∗−b)^2 is equal to a^2−2ab+b^2.

Therefore, we are looking for a term that satisfies the middle term, −40m, when ∗2m is squared.

The desired monomial is 10, yielding the identity (10−2m)^2 = 100−40m+4m^2.

For the second identity (3a+2.5b)^2 = 9a^2+6.25b^2 + ∗, recall that the expansion should have a middle term which is the product of the first and second terms from the original binomial multiplied by 2.

Therefore, we compute 2 × 3a × 2.5b = 15ab, making the identity (3a+2.5b)^2=9a^2+6.25b^2 + 15ab.

Answer 2

To obtain an identity replace * by 10 in first equation and * by 15ab in second equation.

Step-by-step explanation:

Monomial is an algebraic expression consisting of one term only.

Given monomial:

1)
`(*-2m)^2=100-40m+4m^2`

We have to find the value of * in above equation

Using identity
`(a-b)^2=a^2-2ab+b^2`

Comparing with given equation, we get

a= * , b = 2m

then apply the identity, we get

`(*-2m)^2=*^2-2 * 2m * *+(2m)^2`

`\Rightarrow (*-2m)^2=*^2-4m*+4m^2`

now again comparing with the given equation,

`\Rightarrow *^2-4m*+4m^2=100-40m+4m^2`

This gives * = 10 as
`-4m*=-40m \Rightarrow *=10`

2)
`(3a+2.5b)^2=9a^2+6.25b^2 + *`

We have to find the value of * in above equation

Using identity
`(a+b)^2=a^2+2ab+b^2`

Comparing with given equation, we get

a= 3a , b = 2.5b

then apply the identity, we get

`(3a+2.5b)^2=(3a)^2+(2.5b)^2 + 2 * 3a * 2.5b`

`\Rightarrow (3a+2.5b)^2=(3a)^2+(2.5b)^2 +15ab`

now again comparing with the given equation, we get * = 15ab

Thus, to obtain an identity replace * by 10 in first equation and * by 15ab in second equation.