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Perfect square of a number is formed when it is multiplied by itself. Similarly, when a polynomial is multiplied by itself it results in a perfect square polynomial. But it becomes difficult to find the perfect square of sum or difference between two variables. In that situation, we use the Perfect Square Formula that helps in calculating the square of the result obtained after adding or subtracting two variables. In simple words, the Perfect Square Formula finds the square of a binomial.
The formula for finding the Perfect Square of a binomial expression is :
(a ± b)2 = (a2 ± 2ab + b2)
This can be expanded further as two different equations:
(a + b)2 = (a2 + 2ab + b2)
(a - b)2 = (a2 - 2ab + b2)
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Key Takeaways: Perfect Square Formula, Polynomial, Binomial, Perfect Square, Polynomial.
Perfect Square Formula
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Perfect Square Formula is an algebraic expression that calculates the square of a binomial expression. Basically, there are two Perfect Square Formulas, first is when the binomial involves addition of variables and second is when variables are subtracted.
Binomial is any mathematical expression which includes two variables with an addition or subtraction sign between them such as x-2, x+2, x-4 etc.
When variables are added in a binomial expression the formula is as follows:
(a + b)2 = (a2 + 2ab + b2)
When variables are subtracted in binomial expression the formula is as follows:
(a - b)2 = (a2 - 2ab + b2)
Binomial Theorem and Pascal Triangle
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Uses of Perfect Square Formula
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- To find the perfect square of a binomial expression. In higher classes you would also learn about finding the perfect square of trinomials.
- Check whether an expression is a perfect square or not.
- In simplifying a complex equation to a simpler form by factorization.
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Things to Remember
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- Perfect Square is any number which is obtained after multiplying a number by itself.
- The perfect square of a negative number is also positive. For example, 5*5 = 25, where 25 is the perfect square and (-5)*(-5) = 25.
- Perfect Square Formula can also be used in higher polynomials which are discussed in higher classes. For example, (a + b+ c)2 = a2 + b2 + c2 + 2(ab + bc + ca).
- Be careful about the sign change in the Perfect Square Formula. The sign is only changed in front of 2ab.
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Sample Questions
Ques. Verify whether the given expression is a perfect square or not. (2 Marks)
x2 + 12x + 36
We know, The Perfect Square Formula is (a + b)2 = (a2 + 2ab + b2)
Rearrange it in the form of the formula
→ x2 + 2(x.6) + 62
The above equation is in the form of (a + b)2 = (a2 + 2ab + b2) where, a = x, b = 6 and 2ab = 2(x.6)
So, x2 + 2(x.6) + 62 = (x+6)2
Thus, x2 + 12x + 36 is a perfect square.
Ques. Simplify the following binomial expression using The Perfect Square Formula. (3x - 4y)2 (2 Marks)
Ans. Here, a = 3x and b = 4y
Using The Perfect Square Formula, (a - b)2 = (a2 - 2ab + b2)
→ (3x - 4y)2 = {(3x)2 - 2*3x*4y + (4y)2}
= (9x2 - 24xy + 16y2)
Thus, the solution is 9x2 - 24xy + 16y2.
Ques. How to find the perfect square of a higher polynomial where three elements are added such as (a + b + c)2? (1 Mark)
Ans. To find the perfect square of a higher polynomial three elements are added, we need to use the following formula:
(a + b + c)2= a2+b2+c2 + 2(ab +bc+ca)
This is useful in higher classes.
Ques. Apply The Perfect Square Formula to find the solution of (9y - 6)2. (2 Marks)
Ans. Here, a = 9y and b = 6
Applying The Perfect Square Formula on (9y - 6)2 we get,
(9y - 6)2 = (9y)2 - 2*9y*6 + 62
(9y - 6)2 = 81y2 - 108y + 36
Thus, the solution is 81y2 - 108y + 36.
Ques. Verify the following expression for a perfect square. 25y2 + 30xy + 9x2 (2 Marks)
Ans. Rearrange the expression in the form of formula of Perfect Square
We know that, Perfect Square Formula is (a + b)2 = (a2 + 2ab + b2)
= (5y)2 + 2*5y*3x + (3x)2
= (5y + 3x)2, which is a perfect square.
Thus, it is proved that 25y2 + 30xy + 9x2 is a perfect square.
Ques. Find the square of the given binomial expression: (18x - 9y). (2 Marks)
Ans. For finding the square of a binomial expression we use The Perfect Square Formula which states, (a - b)2 = (a2 - 2ab + b2)
Hence, (18x - 9y)2 = {(18x)2 - 2*18x*9y + (9y)2}
(18x - 9y)2 = 324x2 - 324x + 81y2
Thus, the square of the binomial expression, (18x - 9y) is 324x2 - 324x + 81y2.
Ques. If (2x - 3)2 is 16. Then what is the value of 4x2 - 12x + 9? (2 Marks)
Ans. We need to find some relation between (2x - 3)2 and 4x2 - 12x + 9.
By using the perfect square formula,
(2x - 3)2 = (2x)2 - 2*2x*3 + 32
(2x - 3)2 = 4x2 - 12x + 9
Thus, 4x2 - 12x + 9 is the perfect square of (2x - 3)2 .
The given value of (2x - 3)2 is 16. So, the value of 4x2 - 12x + 9 is also 16.
Therefore, 4x2 - 12x + 9 = 16.
Ques. If 16x2 + 24x + 9 is 20, then what is the value of (4x + 3)2? (2 Marks)
Ans. First we need to find some relation between 16x2 + 24x + 9 is 20 and (4x + 3)2.
By rearranging,
→ (4x)2 + 2.4x.3 + (3)2
This is in the form of a2 + 2ab + b2 = (a + b)2 where a = 4x, 2ab = 24x and b = 3
Hence, (4x)2 + 2.4x.3 + (3)2 = (4x + 3)2
Thus, (4x)2 + 2.4x.3 + (3)2 is a perfect square of (4x + 3)2.
Therefore, (4x + 3)2 = (4x)2 + 2.4x.3 + (3)2 = 20.
Ques. What is the definition of identity? (2 Marks)
Ans: An identity is a mathematical equivalence that connects one mathematical expression A to some other mathematical expression B, such that A and B (which may contain some variables) generate the same value for all values of the variables within a given range of validity. In other words, if A and B specify the identical functions, A = B is an identity, but an identity is an equivalence between functions that are specified differently.
Ques. What is the Root of unity? (2 Marks)
Ans: A root of unity, also known as a de Moivre number in mathematics, is any complex number that returns 1 when increased to some positive integer power n. Many disciplines of mathematics use roots of unity, but they're especially significant in number theory, group character theory, and the discrete Fourier transform.
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