a² - b² formula in Algebra is the basic formula in mathematics used to solve various algebraic problems. a² - b² formula is also called the difference of squares formula, as this formula helps us to find the difference between two squares without actually calculating the squares. The image added below shows the formula of a² - b².

It is also used to solve trigonometric, differential, and other problems. This formula tells us that the difference between two square numbers is equal to the product of the sum and difference of the two numbers, i.e.

a² - b² = (a + b).(a - b)

The formula a² - b² allows us to determine the variance between the squares of two numbers without the need to compute the actual square values. The expression for the a² - b² formula is as follows: a² - b² = (a + b).(a - b)

Difference of Squares Formula

The difference of two squares is calculated using the standard algebraic identity a² - b². For example, we are given two variables, a and b, then the difference of their squares is calculated using the formula, a² – b² = (a+b) . (a–b)

Basically, the difference of squares formula says that for any two algebraic variables a and b, the expression a² – b² is equal to the product of the sum and difference of the variables. This identity is used widely to simplify complicated algebraic expressions.

a² - b² Square Formula Proof

a² - b² identity can be proved by simplifying the RHS of the identity. The a² - b² formula is given as,

a² - b² = (a - b)(a + b)

This formula is proved as,

RHS = (a+b) (a–b)
⇒ RHS = a (a–b) + b (a–b)
⇒ RHS = a² – ab + ba – b²
⇒ RHS = a² – ab + ab – b²
⇒ RHS = a² – b²
⇒ RHS = LHS

Hence Proved.

a² + b² Formula

The a² + b² formula is the algebraic formula that is used to find the sum of the squares of two numbers. The sum of the square formula is given as,

a² + b² = (a + b)² - 2ab

The a² + b² formula is used to solve various algebraic problems. Various other important algebraic formulas are added below,

(a + b)² and (a - b)² Formula

The (a + b)² formula is given as,

(a + b)² = a² + b² + 2ab

The (a - b)² formula is given as,

(a - b)² = a² + b² - 2ab

a² - b² Identity

a² - b² identity is one of the algebraic identities that is used to find the difference between the squares of two numbers. This identity has various applications and is given as,

a² - b² = (a - b).(a + b)

All a² + b² Formula

Here are some important and frequently used Algebra formulas involving squares.

• a²– b² = (a – b)(a + b)
• (a + b)²=  a²+ 2ab + b²
• a²+ b²= (a + b)²– 2ab
• (a – b)² = a²– 2ab+ b²
• (a + b + c)² = a² + b² + c²+ 2ab + 2bc + 2ca
• (a – b – c)² = a²+ b²+ c²– 2ab + 2bc – 2ca

Read More,

• Algebra Formula
• Basic Maths Formula
• Algebraic Expression

Solved Question on a² - b² Formula

Question 1: Simplify x² – 16

Solution:

x² – 16
We will use the a² - b² formula to factorize this.
= x² – 4²
We know that, a² – b² = (a+b) (a–b)
Given,

• a = x
• b = 4

= (x + 4)(x – 4)

Question 2: Simplify 9y² – 144

Solution:

9y² – 144
We will use the a² - b² formula to factorize this.
= (3y)² – (12)²
We know that, a² – b² = (a+b)(a–b)
Given,

• a = 3y
• b = 12

= (3y + 12)(3y – 12)

Question 3: Simplify (3x + 2)² – (3x – 2)²

Solution:

We know that,
a² – b² = (a+b)(a–b)

Given,

• a = 3x + 2
• b = 3x – 2

(3x + 2)² – (3x – 2)²
= (3x + 2 + 3x – 2)(3x + 2 – (3x – 2))
= 6x(3x + 2 – 3x + 2)
= 6x(4)
= 24x

Question 4: Simplify y² – 100

Solution:

y² – 100
We will use the a² - b² formula to factorize this.

= y² – (10)²
We know that,
a² – b² = (a+b)(a–b)

Given,

• a = y
• b = 10

= (y + 10)(y – 10)

Question 5: Evaluate (x + 6) (x – 6)

Solution:

We know that,
(a+b) (a–b) = a² – b²

Given,

• a = x
• b = 6

(x + 6) (x – 6)
= x² – 6²
= x² – 36

Question 6: Evaluate (y + 13)(y – 13)

Solution:

We know that,
(a+b) (a–b) = a² – b²

Given,

• a = y
• b = 13

(y + 13).(y – 13)
= y² – (13)²
= y² – 169

Question 7: Evaluate (x + y + z).(x + y – z)

Solution:

We know that,
(a+b) (a–b) = a² – b²

Given,

• a = x + y
• b = z

(x + y + z) (x + y – z)
= (x + y)² – z²
= x² + y² + 2xy – z²

Unsolved Question on (a² - b²) Formula

Q1. Simplify 15² - 14² using a² - b² identity.

Q2. Simplify 11² - 7² using a² - b² identity.

Q3. Solve 23² - 9² using a² - b² identity.

Q4. Solve 9² - 7² using a² - b² identity.