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# (a+b)^2 Formula

The (a + b)^{2} formula is used to find the square of a binomial. This formula is also used to factorize some special types of trinomials and is one of the algebraic identities. The (a + b) whole square formula is the result of the square of the sum of two terms a and b.

The (a + b)^{2} formula is widely used to factorize the trinomials of the form a^{2} + 2ab + b^{2}. This formula is explained below along with solved examples in the following section.

## What is (a+b)^2 Formula?

The (a + b)^{2} formula is the algebraic identity used to find the square of the sum of two numbers. i.e., it is used to find the square of a binomial a + b. The formula of (a+b) whole square says **(a + b) ^{2} = a^{2} + 2ab + b^{2}**. To prove this formula, we will just multiply (a + b (a + b).

(a + b)^{2}= (a + b)(a + b)

= a (a + b) + b (a + b)

= a^{2} + ab + ba + b^{2} (by distributive property)

= a^{2} + 2ab + b^{2} (combined the like terms)

Thus, (a + b)^{2} formula is: (a + b)^{2} = a^{2} + 2ab + b^{2}

## Geometric Proof of a + b Whole Square Formula

To prove the a + b whole square formula geometrically, two squares of lengths 'a' and 'b' are attached as shown in the figure below such that two rectangles, each of area ab are formed. We can understand this formula geometrically using the following figure:

- The length of the big square is (a + b). Thus, its area is (a + b)
^{2}(side × side). - Now, we will calculate the area of the big square by adding up the areas of squares and rectangles which are forming the big square. Then the area of the big square is a
^{2}+ ab + ab + b^{2}which when simplified, we get a^{2}+ 2ab + b^{2}.

Since the above two points represent the area of the same (big) square, we have **(a + b) ^{2} = a^{2} + 2ab + b^{2}**.

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## Examples on (a+b)^2 Formula

**Example 1:** Find the value of (3x + 2y)^{2} using (a + b)^{2} formula.

**Solution:**

To find: The value of (3x + 2y)^{2}.

Let us assume that a = 3x and b = 2y.

We will substitute these values in (a + b)^{2} formula:

(a + b)^{2} = a^{2} + 2ab + b^{2}

(3x + 2y)^{2} =(3x)^{2} + 2(3x)(2y) + (2y)^{2}

= 9x^{2} + 12xy + 4y^{2}

**Answer: (3x + 2y) ^{2} = 9x^{2} + 12xy + 4y^{2}.**

**Example 2:** Factorize x^{2} + 4xy + 4y^{2} using (a + b)^{2} formula.

**Solution:**

To factorize: x^{2} + 4xy + 4y^{2}.

We can write the given expression as: (x)^{2} + 2 (x) (2y) + (2y)^{2}.

Using (a + b)^{2} formula:

a^{2} + 2ab + b^{2} = (a + b)^{2}

Substitute a = x and b = 2y in this formula:

(x)^{2} + 2 (x) (2y) + (2y)^{2}. = (x + 2y)^{2}

**Answer: x ^{2} + 4xy + 4y^{2} = (x + 2y)^{2}.**

**Example 3:** Simplify the following using (a+b)^{2} formula: (7x + 4y)^{2}

**Solution:**

a = 7x and b = 4y

Using formula (a + b)^{2} = a^{2} + 2ab + b^{2}

(7x + 4y)^{2 }= (7x)^{2} + 2(7x)(4y) + (4y)^{2}

= 49x^{2} + 56xy + 16y^{2}

**Answer:** (7x + 4y)^{2} = 49x^{2} + 56xy + 16y^{2}.

## FAQs on (a + b)^{2} Formula

### What Is (a + b)^2?

**(a + b)^2 **is read as "a plus b whole square". Its expansion is given by (a + b)^{2} = a^{2} + 2ab + b^{2}. This can be obtained by simply doing the binomial multiplication (a + b)(a + b).

### What Is the (a + b) Whole Square Formula in Algebra?

The **(a + b) whole square formula** is one of the important algebraic identities. It is pronounced as a plus b whole square. To understand how the formula is derived, we can expand (a + b)^{2} as follows: (a + b)^{2} = (a + b) (a + b) = a^{2} + ab + ba + b^{2} = a^{2} + 2ab + b^{2}.

### What is the Difference Between (a + b) Whole Square and (a - b) Whole Square?

These two formulas almost look like same, except for one difference, the sign of the term 2ab.

- (a + b)
^{2}= a^{2}+ 2ab + b^{2} - (a - b)
^{2}= a^{2}- 2ab + b^{2}

### How To Apply the (a + b)^2 Formula to Simplify Numbers?

Let us understand the use of the (a + b)^{2} formula with the help of the following example.

**Example:** Find the value of 25^{2} using the (a + b)^{2} formula.

**Solution:**

We know that 25 = 20 + 5. Thus, 25^{2} = (20+5)^{2}.

Let us assume that a = 20 and b = 5.

We will substitute these in the formula of (a + b)^{2}.

(a + b)^{2} = a^{2} + 2ab + b^{2}

25^{2}= (20+5)^{2}

= 20^{2} + 2(20)(5) + 5^{2}

= 400 + 200 + 25

= 625

**Answer:** 25^{2} = 625.

### How To Use the (a + b)^2 Formula?

The following steps are followed while using (a + b)^{2} formula.

- To begin with, observe the pattern of the numbers whether the numbers have whole ^2 as power or not.
- Write down the formula of (a + b)
^{2}. - (a + b)
^{2}= a^{2}+ 2ab + b^{2}. - Substitute the values of a and b in the (a + b)
^{2}formula and simplify.

### How is (a + b)^2 Formula Useful in Algebraic Equations?

The (a + b) whole square formula is useful while solving algebraic equations to expand the brackets with binomials.

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