Identities：練習問題

Which of the following terms correctly completes the expression \( 9x^2 + \_\_\_ + 25y^2 \) to form a perfect square trinomial?

If \(x + \frac{1}{x} = 4\), what is the value of \(x^2 + \frac{1}{x^2}\)?

Given that the following is an identity in \(x\):

\(A(x - 1)(x - 3) + B(x - 1) + C \equiv x^2 - 6x + 10\)

Find the value of \(A + B + C\).

Expand the algebraic expression: \((x+1)^2\)

If \(x+y = 7\) and \(x^2+y^2 = 29\), what is the value of \(xy\)?

Factorise the quadratic expression \(z^2 - 14z + 49\) using a standard identity.

If \(a^2 + b^2 = 50\) and \(ab = 7\), find the value of \((a-b)^2\) without finding the values of \(a\) and \(b\).

Prove the following identity:
\((x + 2y)^2 - (x - 2y)^2 = 8xy\)

Expand and simplify the expression \( (2a+3)^2 - (4a^2 - 9) \).

Expand and simplify the expression: $$(2x + 5)^2 - (2x - 5)(2x + 5)$$