Question

65.c
insert a monomial into each trinomial such that the result is a perfect square.
$25b^2 + 70b + ....$
answer
$25b^2 + 70b + \boxed{\\x}$

Explanation:

Step1: Recall perfect square formula

The perfect square trinomial formula is \((a + b)^2 = a^2 + 2ab + b^2\). For the given trinomial \(25b^2+70b+\dots\), we can see that \(25b^2=(5b)^2\), so \(a = 5b\).

Step2: Find the middle term relation

The middle term of the perfect square trinomial is \(2ab\). Here, the middle term is \(70b\), and \(a = 5b\), so \(2ab=2\times5b\times b'=70b\) (let the missing term be \(b'^2\)). Solving for \(b'\): \(2\times5b\times b' = 70b\), divide both sides by \(10b\) (assuming \(b
eq0\)), we get \(b' = 7\).

Step3: Find the missing monomial

The missing term (the last term of the perfect square trinomial) is \(b'^2\), so \(b'^2 = 7^2=49\). So the trinomial becomes \((5b + 7)^2=25b^2+70b + 49\).

Answer:

\(49\)