Question

insert terms to make identy
(3a+2.5b)^2=9a^2+6.25b^2+___

Answer 1

To make the equation \( (3a + 2.5b)^2 = 9a^2 + 6.25b^2 + \_\_\_ \) an identity, we need to expand the left-hand side.

The square of a binomial \( (x+y)^2 \) is given by \( x^2 + 2xy + y^2 \).

Here, let's identify \( x \) and \( y \):

- \( x = 3a \)
- \( y = 2.5b \)

Now, applying the formula:

\[
(3a + 2.5b)^2 = (3a)^2 + 2(3a)(2.5b) + (2.5b)^2
\]

Calculating each term:

1. \( (3a)^2 = 9a^2 \)
2. \( 2(3a)(2.5b) = 2 \cdot 3 \cdot 2.5 \cdot ab = 15ab \)
3. \( (2.5b)^2 = 6.25b^2 \)

Now combine these results:

\[
(3a + 2.5b)^2 = 9a^2 + 15ab + 6.25b^2
\]

Now we can rewrite the original equation with the addition:

\[
(3a + 2.5b)^2 = 9a^2 + 6.25b^2 + 15ab
\]

Thus, the blank in the equation is filled by:

\[
15ab
\]

Therefore, the completed identity is:

\[
(3a + 2.5b)^2 = 9a^2 + 6.25b^2 + 15ab
\]