Question

Replace ∗ with a monomial so that the expression can be rewritten as a square of a sum or a difference:

25a^2+ ∗ + 1/4b^2

Answer 1

`\( 25a^2 + (1)/(4)b^2 = (5a + (1)/(2)b)^2 \)`

Step-by-step explanation:

In the given expression,
`\(25a^2 + (1)/(4)b^2\),` we need to find a monomial to replace the ∗ so that the expression can be expressed as a square of a sum or a difference. To achieve this, we recognize that
`\(25a^2\)` is a perfect square, as is
`\((1)/(4)b^2\)`.

The square root of
`\(25a^2\)` is
`\(5a\)`, and the square root of
`\((1)/(4)b^2\) is \((1)/(2)b\)`. Therefore, by choosing the monomial that combines these square roots, we get
`\( (5a + (1)/(2)b)^2 \).`

This works because when you expand
`\( (5a + (1)/(2)b)^2 \)`, you get the original expression
`\(25a^2 + (1)/(4)b^2\)`. The process of factoring and expanding squares is a fundamental algebraic technique, and recognizing perfect squares allows for simplifying expressions and making them more manageable.

In this case, understanding the square roots of the terms in the expression helps to rewrite it in a more concise and recognizable form.