Applications of Identities

Here are four problems using algebraic identities with detailed solutions:

Problem 1: Expand and verify if 2x−52 = 4x2 - 20x + 25 is an identity.

Solution: Using the identity a−b2 = a2 - 2ab + b2. Let a = 2x and b =

2x−52 = 2x2 - 2(2x)(5) + 52 = 4x²9² - +

Since RHS = LHS for any value of x, this is an .

Let's verify for x = 2:

LHS = 4−52 = −12 =

RHS = 4(4) - 20(2) + 25 = - + =

Problem 2: Simplify: (x + 3)(x - 3) + 9

Solution:

Using the identity (a + b)(a - b) = a2 - b2

(x + 3)(x - 3) + 9 = x22x2 - + = x²1

This means (x + 3)(x - 3) + 9 ≡ x2

Problem 3: Expand 3x+2y2

Solution:

Using the identity a+b2 = a2 + 2ab + b2

Let a = 3x and b =

3x+2y2 = 3x2 + 2(3x)(2y) + 2y2 = 9x28x2 + + 4y26y2

Therefore, it is an identity as LHS = RHS for all values of x and y.

Each of these problems demonstrates different aspects of algebraic identities and how they can be used to simplify or verify expressions. They show that identities remain true regardless of the values substituted for the variables.