Pythagorean theorem

     

Thea2- b2formulais also known as "the difference of squaresformula".The a square minus b square is used lớn find the differencebetween the two squareswithout actually calculating the squares.

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It is used khổng lồ factorize the binomials of squares.

What is a^2-b^2Formula?

The a2- b2formula is given as: a2- b2= (a -b) (a + b)

If you would like to verify this, you can justmultiply(a - b) (a + b) và see whether you get a2- b2.

Verification of a2- b2Formula

Let us see the proof of a square minus b square formula. Lớn verify thata2- b2= (a -b) (a + b) we need to prove LHS = RHS. Let us try lớn solve the equation:

a2- b2= (a -b) (a + b)Multiply(a -b) and(a +b) we get=a(a+b) -b(a + b)=a2 + ab - ba - b2=a2 + 0 + b2=a2- b2Hence Verifieda2- b2= (a -b) (a + b)

You can understand the a2- b2formula geometrically using the following figure:

*

Proof of a^2 - b^2 Formula

The proof thatthe value a2- b2is(a + b)(a - b). Let us consider the above figure. Take the two squares of sides a units và b units respectively. This can also be represented as the sum of are areas of two rectangles as presented in the below figure.

Onerectanglehas a length of a unit và a breadth of (a - b)units on the other side the second rectangle hasa length of (a - b) and a breadth of b units. Now địa chỉ the areas of the two rectangles to obtain the resultant values. The respective areas of the two rectangles are (a -b)× a = a(a -b) , & (a - b)× b = b(a - b). The sum of theareas of rectangles is the actualobtainedresultant expression i.e., a(a + b) + b(a - b) = (a + b)(a - b). Again re-arranging the individualrectangles and squares, we get: (a+b)(a−b)=a2−b2



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Examples ona^2-b^2 Formula

Let us solve some interesting problems using the a^2-b^2 formula.

Example 1: Using a2- b2formula find the value of 1062- 62.

Solution:To find:1002- 62.

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Let us assume that a = 100 & b = 6.We will substitute these in the a2- b2formula.a2- b2= (a -b) (a + b)1062- 62= (106 - 6) (106 + 6)= (100) (112)= 11200

Answer:1062- 62= 11200.

Example 2:Factorize the expression 25x2- 64.

Solution:To factorize: 25x2- 64.We will use thea2- b2formula khổng lồ factorize this.We can write the given expression as25x2- 64= (5x)2- 82We will substitute a = 5xand b = 8in the formula ofa2- b2.a2- b2= (a -b) (a + b)(5x)2- 82= (5x - 8) (5x + 8)

Answer: 25x2- 64= (5x - 8) (5x + 8)

Example 3:Simplify 102- 52usinga2-b2formula

Solution:To find102- 52

Let us assume a = 10 & b = 5Using formulaa2- b2= (a - b) (a+ b)102-52= (10 - 5) (10 + 5)= 10(10 +5) - 5(10 + 5)= 10(15) - 5(15)= 150-75 = 75Answer:102-52= 75.


FAQs on a^2-b^2Formula

What Is the Expansion of a2-b2Formula?

a2-b2formula is read as a squareminus b square. Its expansion is expressed asa2-b2= (a -b) (a + b)

What Is thea2-b2Formula in Algebra?

The a2-b2formula is also known as one of the importantalgebraic identity. It is read as a squareminus b square. Its a2-b2formula is expressed asa2-b2= (a -b) (a+ b)

How lớn Simplify Numbers Usingthea2-b2Formula?

Let us understand the use of the a2-b2formula with the help of the following example.Example:Find the value of 102- 22using the a2-b2formula.To find:102-22Let us assume that a = 10and b = 2.We will substitute these in the formula ofa2- b2.a2- b2= (a -b) (a+ b)102-22= (10-2)(10+ 2)= 10 (10 + 2) - 2 (10 + 2)= 10(12) - 2(12)=120 - 24 = 96Answer:102-22= 96.

How khổng lồ Use thea2-b2Formula Give Steps?

The following steps are followed while usinga2-b2formula.

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Firstlyobserve the pattern of the numbers whether thenumbers have ^2 as power or not.Write down the formula ofa2- b2a2- b2= (a -b) (a+ b)Substitute the values of a and b in thea2-b2formula & simplify.