OP Malhotra Class 9 Maths Solutions Chapter 3 Expansions Exercise 3 (A)

Question 1. Write down the products for each of the following:
(i) (x + 4) (x + 2)
(ii) (4a – 5) (5a + 6)
(iii) (xy + 6) (xy – 5)
(iv) (7x² – 5y) (x² – 3y)

Answer:
(i) We use the identity \( (a+b)(a+c) = a^2 + (b+c)a + bc \).
\( (x + 4) (x + 2) = x^2 + (4 + 2)x + (4 \times 2) \)
\( = x^2 + 6x + 8 \)
(ii) We multiply each term carefully.
\( (4a - 5) (5a + 6) = 4a \times 5a + 4a \times 6 - 5 \times 5a - 5 \times 6 \)
\( = 20a^2 + 24a - 25a - 30 \)
\( = 20a^2 - a - 30 \)
(iii) We use the identity \( (y+b)(y+c) = y^2 + (b+c)y + bc \) where \( y=xy \).
\( (xy + 6) (xy - 5) = (xy)^2 + (6 - 5)xy + (6 \times -5) \)
\( = x^2y^2 + xy - 30 \)
(iv) We distribute each term in the first expression by each term in the second expression.
\( (7x^2 - 5y) (x^2 - 3y) = 7x^2 \times x^2 - 7x^2 \times 3y - 5y \times x^2 + 5y \times 3y \)
\( = 7x^4 - 21x^2y - 5x^2y + 15y^2 \)
\( = 7x^4 - 26x^2y + 15y^2 \)
In simple words: To find the product, multiply each part of the first bracket by each part of the second bracket. Then, combine any terms that are similar. Remember to follow the rules of signs.

🎯 Exam Tip: Pay close attention to negative signs when multiplying terms, especially when combining like terms. A common mistake is miscalculating terms like \( -5 \times 6 \) or \( 6 \times -5 \).

Question 2. Write down the squares of the following expressions
(i) 3x + 5y
(ii) 5y – 2z
(iii) \( 5p - \frac { 1 }{ 4q } \)
(iv) (5x + 3y + z)²
(v) (- 3m – 5n + 2p)²
(vi) \( (2x - \frac { 1 }{ 3 }p + 3q)² \)

Answer:
(i) We use the identity \( (A+B)^2 = A^2 + 2AB + B^2 \).
\( (3x + 5y)^2 = (3x)^2 + 2 \times (3x) \times (5y) + (5y)^2 \)
\( = 9x^2 + 30xy + 25y^2 \)
(ii) We use the identity \( (A-B)^2 = A^2 - 2AB + B^2 \).
\( (5y - 2z)^2 = (5y)^2 - 2 \times (5y) \times (2z) + (2z)^2 \)
\( = 25y^2 - 20yz + 4z^2 \)
(iii) We use the identity \( (A-B)^2 = A^2 - 2AB + B^2 \).
\( \left(5p - \frac{1}{4q}\right)^2 = (5p)^2 - 2 \times (5p) \times \left(\frac{1}{4q}\right) + \left(\frac{1}{4q}\right)^2 \)
\( = 25p^2 - \frac{10p}{4q} + \frac{1}{16q^2} \)
\( = 25p^2 - \frac{5p}{2q} + \frac{1}{16q^2} \)
(iv) We use the identity \( (A+B+C)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \).
\( (5x + 3y + z)^2 = (5x)^2 + (3y)^2 + (z)^2 + 2(5x)(3y) + 2(3y)(z) + 2(z)(5x) \)
\( = 25x^2 + 9y^2 + z^2 + 30xy + 6yz + 10zx \)
(v) We use the identity \( (A+B+C)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \).
\( (-3m - 5n + 2p)^2 = (-3m)^2 + (-5n)^2 + (2p)^2 + 2(-3m)(-5n) + 2(-5n)(2p) + 2(2p)(-3m) \)
\( = 9m^2 + 25n^2 + 4p^2 + 30mn - 20np - 12pm \)
(vi) We use the identity \( (A+B+C)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \).
\( \left(2x - \frac{1}{3}p + 3q\right)^2 = (2x)^2 + \left(-\frac{1}{3}p\right)^2 + (3q)^2 + 2(2x)\left(-\frac{1}{3}p\right) + 2\left(-\frac{1}{3}p\right)(3q) + 2(3q)(2x) \)
\( = 4x^2 + \frac{1}{9}p^2 + 9q^2 - \frac{4}{3}xp - 2pq + 12qx \)
In simple words: To square an expression, multiply it by itself. If there are two terms, use the formula \( (A+B)^2 \) or \( (A-B)^2 \). If there are three terms, use the formula \( (A+B+C)^2 \). Remember to square each term and include the cross-product terms.

🎯 Exam Tip: Be very careful with the signs when expanding squares. A negative number squared always becomes positive, but a product of two terms might remain negative if one of the terms is negative.

Question 3. Simplify:
\( (2x - p + c)^2 - (2x + p - c)^2 \)

Answer:
We can use the identity \( A^2 - B^2 = (A-B)(A+B) \).
Here, let \( A = (2x - p + c) \) and \( B = (2x + p - c) \).
\( A - B = (2x - p + c) - (2x + p - c) = 2x - p + c - 2x - p + c = -2p + 2c \)
\( A + B = (2x - p + c) + (2x + p - c) = 2x - p + c + 2x + p - c = 4x \)
So, \( (2x - p + c)^2 - (2x + p - c)^2 = (-2p + 2c)(4x) \)
\( = 4x(-2p + 2c) \)
\( = -8xp + 8cx \)
Alternatively, expand each square separately using \( (A+B+C)^2 \) formula and then subtract.
\( (2x - p + c)^2 = (2x)^2 + (-p)^2 + c^2 + 2(2x)(-p) + 2(-p)(c) + 2(c)(2x) \)
\( = 4x^2 + p^2 + c^2 - 4xp - 2pc + 4cx \)
\( (2x + p - c)^2 = (2x)^2 + p^2 + (-c)^2 + 2(2x)(p) + 2(p)(-c) + 2(-c)(2x) \)
\( = 4x^2 + p^2 + c^2 + 4xp - 2pc - 4cx \)
Now, subtract the second expansion from the first:
\( (4x^2 + p^2 + c^2 - 4xp - 2pc + 4cx) - (4x^2 + p^2 + c^2 + 4xp - 2pc - 4cx) \)
\( = 4x^2 + p^2 + c^2 - 4xp - 2pc + 4cx - 4x^2 - p^2 - c^2 - 4xp + 2pc + 4cx \)
\( = (-4xp - 4xp) + (-2pc + 2pc) + (4cx + 4cx) + (4x^2 - 4x^2) + (p^2 - p^2) + (c^2 - c^2) \)
\( = -8xp + 8cx \)
In simple words: This problem asks us to find the difference between two squared expressions. We can solve it by using the special formula \( A^2 - B^2 = (A-B)(A+B) \), which is often quicker than expanding each square fully. We substitute the terms into this formula and then multiply the results.

🎯 Exam Tip: When simplifying expressions like \( A^2 - B^2 \), always consider using the difference of squares identity \( (A-B)(A+B) \). It can significantly reduce the amount of calculation needed and minimize errors compared to expanding each square separately.

Question 4. Write down the following products :
(i) (3b + 7) (3b – 7)
(ii) \( \left(\frac{1}{3}-5 x\right)\left(\frac{1}{3}+5 x\right) \)
(iii) (x³ – 3) (x³ + 3)
(iv) \( \left(a^4-\frac{1}{5 y}\right)\left(a^4+\frac{1}{5 y}\right) \)

Answer:
We use the identity \( (a + b) (a - b) = a^2 - b^2 \) for all parts.
(i) Here, \( a = 3b \) and \( b = 7 \).
\( (3b + 7) (3b - 7) = (3b)^2 - (7)^2 \)
\( = 9b^2 - 49 \)
(ii) Here, \( a = \frac{1}{3} \) and \( b = 5x \).
\( \left(\frac{1}{3} - 5x\right)\left(\frac{1}{3} + 5x\right) = \left(\frac{1}{3}\right)^2 - (5x)^2 \)
\( = \frac{1}{9} - 25x^2 \)
(iii) Here, \( a = x^3 \) and \( b = 3 \).
\( (x^3 - 3) (x^3 + 3) = (x^3)^2 - (3)^2 \)
\( = x^{3 \times 2} - 9 \)
\( = x^6 - 9 \)
(iv) Here, \( a = a^4 \) and \( b = \frac{1}{5y} \).
\( \left(a^4 - \frac{1}{5y}\right)\left(a^4 + \frac{1}{5y}\right) = (a^4)^2 - \left(\frac{1}{5y}\right)^2 \)
\( = a^8 - \frac{1}{25y^2} \)
In simple words: For these problems, we use a special formula called "difference of squares". It says that if you multiply \( (A+B) \) by \( (A-B) \), you always get \( A^2 - B^2 \). Just find your 'A' and 'B' parts and then square them and subtract.

🎯 Exam Tip: Recognize the "difference of squares" pattern \( (a+b)(a-b) = a^2 - b^2 \) immediately. Applying this identity correctly is faster and less prone to errors than full expansion.

Question 5. Find the products :
(i) (x + y) (x - y) (x² + y²)
(ii) (a² + b²) (a⁴ + b⁴) (a + b) (a – b)

Answer:
(i) First, multiply the terms that fit the difference of squares formula.
\( (x + y) (x - y) (x^2 + y^2) \)
\( = [(x)^2 - (y)^2] (x^2 + y^2) \)
\( = (x^2 - y^2) (x^2 + y^2) \)
Now, apply the difference of squares formula again.
\( = (x^2)^2 - (y^2)^2 \)
\( = x^4 - y^4 \)
(ii) Rearrange the terms to group factors that fit the difference of squares formula.
\( (a^2 + b^2) (a^4 + b^4) (a + b) (a - b) \)
\( = (a^2 + b^2) (a^4 + b^4) [(a)^2 - (b)^2] \)
\( = (a^2 + b^2) (a^4 + b^4) (a^2 - b^2) \)
Rearrange again to group for the difference of squares.
\( = (a^2 - b^2) (a^2 + b^2) (a^4 + b^4) \)
\( = [(a^2)^2 - (b^2)^2] (a^4 + b^4) \)
\( = (a^4 - b^4) (a^4 + b^4) \)
Apply the formula one last time.
\( = (a^4)^2 - (b^4)^2 \)
\( = a^8 - b^8 \)
In simple words: For these problems, we multiply the terms in a special order. We look for pairs that look like \( (A+B)(A-B) \) and turn them into \( A^2-B^2 \). We keep doing this until all parts are multiplied. This helps simplify complex multiplications.

🎯 Exam Tip: When you see multiple factors, look for pairs that can be simplified using the difference of squares identity. Work step-by-step, applying the identity repeatedly until the expression is fully multiplied, making sure to group terms correctly.

Question 6. State which of the following expressions is a perfect square :
(i) x² + 8x + 16
(ii) y² + 3y + 9
(iii) 4m² + 4m + 1
(iv) \( 4x^2 - 2 + \frac { 1 }{ 4x^2 } \)
(v) m² – 6m + 4

Answer:
An expression is a perfect square if it can be written in the form \( (a+b)^2 \) or \( (a-b)^2 \).
(i) Consider the expression \( x^2 + 8x + 16 \).
We can write this as \( (x)^2 + 2 \times x \times 4 + (4)^2 \).
This matches the form \( a^2 + 2ab + b^2 = (a+b)^2 \), where \( a=x \) and \( b=4 \).
So, \( x^2 + 8x + 16 = (x + 4)^2 \).
Therefore, it is a perfect square of \( (x + 4) \).
(ii) Consider the expression \( y^2 + 3y + 9 \).
We can write this as \( (y)^2 + 3y + (3)^2 \).
For it to be a perfect square, the middle term \( 3y \) should be \( 2 \times y \times 3 = 6y \).
Since \( 3y \) is not equal to \( 6y \), this expression is not a perfect square.
(iii) Consider the expression \( 4m^2 + 4m + 1 \).
We can write this as \( (2m)^2 + 2 \times (2m) \times 1 + (1)^2 \).
This matches the form \( a^2 + 2ab + b^2 = (a+b)^2 \), where \( a=2m \) and \( b=1 \).
So, \( 4m^2 + 4m + 1 = (2m + 1)^2 \).
Therefore, it is a perfect square of \( (2m + 1) \).
(iv) Consider the expression \( 4x^2 - 2 + \frac { 1 }{ 4x^2 } \).
We can write this as \( (2x)^2 - 2 \times (2x) \times \left(\frac{1}{2x}\right) + \left(\frac{1}{2x}\right)^2 \).
This matches the form \( a^2 - 2ab + b^2 = (a-b)^2 \), where \( a=2x \) and \( b=\frac{1}{2x} \).
So, \( 4x^2 - 2 + \frac { 1 }{ 4x^2 } = \left(2x - \frac{1}{2x}\right)^2 \).
Therefore, it is a perfect square of \( \left(2x - \frac{1}{2x}\right) \).
(v) Consider the expression \( m^2 - 6m + 4 \).
We can write this as \( (m)^2 - 6m + (2)^2 \).
For it to be a perfect square, the middle term \( -6m \) should be \( -2 \times m \times 2 = -4m \).
Since \( -6m \) is not equal to \( -4m \), this expression is not a perfect square.
In simple words: A perfect square is a number or expression that can be written as another number or expression multiplied by itself. To check if something is a perfect square, we look if it fits the pattern \( (A+B)^2 \) or \( (A-B)^2 \). This means checking if the first and last terms are squares and if the middle term is exactly twice the product of their square roots.

🎯 Exam Tip: To identify a perfect square trinomial \( ax^2 + bx + c \), check if \( a \) and \( c \) are perfect squares and if \( b = \pm 2\sqrt{a}\sqrt{c} \). For binomials, it's about matching the \( (A \pm B)^2 \) formula components.

Question 7. If 4x² – 12x + k is a perfect square, find the numerical value of k.

Answer:
For the expression \( 4x^2 - 12x + k \) to be a perfect square, it must be in the form \( (ax - b)^2 \) or \( (ax + b)^2 \).
Comparing with \( (ax - b)^2 = a^2x^2 - 2abx + b^2 \).
Here, \( a^2x^2 = 4x^2 \implies (2x)^2 \), so \( ax = 2x \). This means \( a=2 \).
The middle term is \( -12x \). From the formula, the middle term is \( -2abx \).
So, \( -2(2)(b)x = -12x \)
\( -4bx = -12x \)
\( 4b = 12 \)
\( b = 3 \)
The last term in the perfect square formula is \( b^2 \).
Therefore, \( k = b^2 = (3)^2 \)
\( k = 9 \)
So, the numerical value of \( k \) is 9.
In simple words: To make the expression a perfect square, we need to find a number \( k \) that fits the pattern \( (something)^2 \). We look at the first term to find the 'something' and then use the middle term to figure out the rest. For \( 4x^2 - 12x + k \), it must be like \( (2x - 3)^2 \), so \( k \) must be \( 3^2 \), which is 9.

🎯 Exam Tip: When finding a missing term to complete a perfect square, always identify the 'a' and 'b' terms from the given parts of the expression and use the \( (a \pm b)^2 \) formula to find the missing square term.

Question 8. What term should be added to each of the following expression to make it a perfect square?
(i) 4a² + 28a
(ii) 36a² + 49b²
(iii) 4a² + 81
(iv) 9a² + 2ab + b²
(v) 49a⁴ + 50a²b² + 16b⁴

Answer:
(i) Consider \( 4a^2 + 28a \).
We want to make this \( (2a + b)^2 = (2a)^2 + 2(2a)b + b^2 = 4a^2 + 4ab + b^2 \).
Comparing \( 4a^2 + 28a \) with \( 4a^2 + 4ab \):
\( 4ab = 28a \)
\( 4b = 28 \)
\( b = 7 \)
So, the term to be added is \( b^2 = (7)^2 = 49 \).
Adding 49 makes it \( (2a + 7)^2 \).
(ii) Consider \( 36a^2 + 49b^2 \).
We want to make this \( (6a + 7b)^2 = (6a)^2 + 2(6a)(7b) + (7b)^2 = 36a^2 + 84ab + 49b^2 \).
The missing middle term is \( 2(6a)(7b) = 84ab \).
So, the term to be added is \( 84ab \).
Adding \( 84ab \) makes it \( (6a + 7b)^2 \).
(iii) Consider \( 4a^2 + 81 \).
We want to make this \( (2a + 9)^2 = (2a)^2 + 2(2a)(9) + (9)^2 = 4a^2 + 36a + 81 \).
The missing middle term is \( 2(2a)(9) = 36a \).
So, the term to be added is \( 36a \).
Adding \( 36a \) makes it \( (2a + 9)^2 \).
(iv) Consider \( 9a^2 + 2ab + b^2 \).
We want to make this \( (3a + b)^2 = (3a)^2 + 2(3a)(b) + b^2 = 9a^2 + 6ab + b^2 \).
The given expression is \( 9a^2 + 2ab + b^2 \).
To change \( 2ab \) to \( 6ab \), we need to add \( 6ab - 2ab = 4ab \).
So, the term to be added is \( 4ab \).
Adding \( 4ab \) makes it \( (3a + b)^2 \).
(v) Consider \( 49a^4 + 50a^2b^2 + 16b^4 \).
We want to make this \( (7a^2 + 4b^2)^2 = (7a^2)^2 + 2(7a^2)(4b^2) + (4b^2)^2 \)
\( = 49a^4 + 56a^2b^2 + 16b^4 \).
The given middle term is \( 50a^2b^2 \). The required middle term is \( 56a^2b^2 \).
To change \( 50a^2b^2 \) to \( 56a^2b^2 \), we need to add \( 56a^2b^2 - 50a^2b^2 = 6a^2b^2 \).
So, the term to be added is \( 6a^2b^2 \).
Adding \( 6a^2b^2 \) makes it \( (7a^2 + 4b^2)^2 \).
In simple words: To make an expression a perfect square, we need to add a specific term. We look at the first and last parts of the expression and imagine them as \( A^2 \) and \( B^2 \). Then we find the middle term \( 2AB \) that is needed to complete the square. We add the difference between the needed term and the term already present.

🎯 Exam Tip: Always identify the components \( A \) and \( B \) from the terms provided, then calculate the required middle term \( 2AB \). If a middle term is already present, subtract it from the required \( 2AB \) to find the exact term to be added.

Question 9. Write down the expansion of the following
(i) (a + 1)³
(ii) (3x – 2y)³
(iii) (x² + y)³
(iv) \( (2x - \frac { 1 }{ 3x })³ \)
(v) \( \left(\frac{a}{5}+\frac{b}{2}\right)^3 \)

Answer:
(i) We use the identity \( (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3 \).
\( (a + 1)^3 = (a)^3 + 3(a)^2(1) + 3(a)(1)^2 + (1)^3 \)
\( = a^3 + 3a^2 + 3a + 1 \)
(ii) We use the identity \( (A-B)^3 = A^3 - 3A^2B + 3AB^2 - B^3 \).
\( (3x - 2y)^3 = (3x)^3 - 3(3x)^2(2y) + 3(3x)(2y)^2 - (2y)^3 \)
\( = 27x^3 - 3(9x^2)(2y) + 3(3x)(4y^2) - 8y^3 \)
\( = 27x^3 - 54x^2y + 36xy^2 - 8y^3 \)
(iii) We use the identity \( (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3 \).
\( (x^2 + y)^3 = (x^2)^3 + 3(x^2)^2(y) + 3(x^2)(y)^2 + (y)^3 \)
\( = x^{2 \times 3} + 3x^4y + 3x^2y^2 + y^3 \)
\( = x^6 + 3x^4y + 3x^2y^2 + y^3 \)
(iv) We use the identity \( (A-B)^3 = A^3 - 3A^2B + 3AB^2 - B^3 \).
\( \left(2x - \frac{1}{3x}\right)^3 = (2x)^3 - 3(2x)^2\left(\frac{1}{3x}\right) + 3(2x)\left(\frac{1}{3x}\right)^2 - \left(\frac{1}{3x}\right)^3 \)
\( = 8x^3 - 3(4x^2)\left(\frac{1}{3x}\right) + 3(2x)\left(\frac{1}{9x^2}\right) - \frac{1}{27x^3} \)
\( = 8x^3 - \frac{12x^2}{3x} + \frac{6x}{9x^2} - \frac{1}{27x^3} \)
\( = 8x^3 - 4x + \frac{2}{3x} - \frac{1}{27x^3} \)
(v) We use the identity \( (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3 \).
\( \left(\frac{a}{5}+\frac{b}{2}\right)^3 = \left(\frac{a}{5}\right)^3 + 3\left(\frac{a}{5}\right)^2\left(\frac{b}{2}\right) + 3\left(\frac{a}{5}\right)\left(\frac{b}{2}\right)^2 + \left(\frac{b}{2}\right)^3 \)
\( = \frac{a^3}{125} + 3\left(\frac{a^2}{25}\right)\left(\frac{b}{2}\right) + 3\left(\frac{a}{5}\right)\left(\frac{b^2}{4}\right) + \frac{b^3}{8} \)
\( = \frac{a^3}{125} + \frac{3a^2b}{50} + \frac{3ab^2}{20} + \frac{b^3}{8} \)
In simple words: To expand an expression raised to the power of three, we use a specific algebraic formula. If the terms are added, we use \( (A+B)^3 \). If they are subtracted, we use \( (A-B)^3 \). These formulas help us find all the parts when the expression is fully multiplied out.

🎯 Exam Tip: Memorize the binomial expansion formulas for cubes: \( (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3 \) and \( (A-B)^3 = A^3 - 3A^2B + 3AB^2 - B^3 \). Applying these correctly, especially with fractions and negative signs, is key to getting full marks.