Algebraic Identities

Q1: Expand:

i.  \((x+3)^2\)

Step 1: Here, \(a = x, b = 3\).
Step 2: Apply identity:
\((x+3)^2 = x^2 + 2(x)(3) + 3^2\).
Step 3: Simplify:
\(= x^2 + 6x + 9\).
Answer: \(x^2 + 6x + 9\)

ii.  \((2a+7)^2\)

Step 1: Here, \(a = 2a, b = 7\).
Step 2: Apply identity:
\((2a+7)^2 = (2a)^2 + 2(2a)(7) + 7^2\).
Step 3: Simplify:
\(= 4a^2 + 28a + 49\).
Answer: \(4a^2 + 28a + 49\)

iii.  \((8+3p)^2\)

Step 1: Here, \(a = 8, b = 3p\).
Step 2: Apply identity:
\((8+3p)^2 = 8^2 + 2(8)(3p) + (3p)^2\).
Step 3: Simplify:
\(= 64 + 48p + 9p^2\).
Answer: \(9p^2 + 48p + 64\)

iv.  \((\sqrt3x+2)^2\)

Step 1: Here, \(a = \sqrt3x, b = 2\).
Step 2: Apply identity:
\((\sqrt3x+2)^2 = (\sqrt3x)^2 + 2(\sqrt3x)(2) + 2^2\).
Step 3: Simplify:
\(= 3x^2 + 4\sqrt3x + 4\).
Answer: \(3x^2 + 4\sqrt3x + 4\)

v.  \((4+\sqrt5y)^2\)

Step 1: Here, \(a = 4, b = \sqrt5y\).
Step 2: Apply identity:
\((4+\sqrt5y)^2 = 4^2 + 2(4)(\sqrt5y) + (\sqrt5y)^2\).
Step 3: Simplify:
\(= 16 + 8\sqrt5y + 5y^2\).
Answer: \(5y^2 + 8\sqrt5y + 16\)

vi.  \((6x+11y)^2\)

Step 1: Here, \(a = 6x, b = 11y\).
Step 2: Apply identity:
\((6x+11y)^2 = (6x)^2 + 2(6x)(11y) + (11y)^2\).
Step 3: Simplify:
\(= 36x^2 + 132xy + 121y^2\).
Answer: \(36x^2 + 132xy + 121y^2\)

vii.  \(\left(\frac{x}{2}+\frac{y}{3}\right)^2\)

Step 1: Here, \(a = \frac{x}{2}, b = \frac{y}{3}\).
Step 2: Apply identity:
\(\left(\frac{x}{2}+\frac{y}{3}\right)^2 = \left(\frac{x}{2}\right)^2 + 2\left(\frac{x}{2}\right)\left(\frac{y}{3}\right) + \left(\frac{y}{3}\right)^2\).
Step 3: Simplify:
\(= \frac{x^2}{4} + \frac{xy}{3} + \frac{y^2}{9}\).
Answer: \(\frac{x^2}{4} + \frac{xy}{3} + \frac{y^2}{9}\)

viii.  \(\left(\frac{3a}{5}+\frac{5b}{3}\right)^2\)

Step 1: Here, \(a = \frac{3a}{5}, b = \frac{5b}{3}\).
Step 2: Apply identity:
\(\left(\frac{3a}{5}+\frac{5b}{3}\right)^2 = \left(\frac{3a}{5}\right)^2 + 2\left(\frac{3a}{5}\right)\left(\frac{5b}{3}\right) + \left(\frac{5b}{3}\right)^2\).
Step 3: Simplify:
\(= \frac{9a^2}{25} + 2ab + \frac{25b^2}{9}\).
Answer: \(\frac{9a^2}{25} + 2ab + \frac{25b^2}{9}\)

Q2: Expand:

i.  \((x-9)^2\)

Step 1: Here, \(a = x, b = 9\).
Step 2: Apply identity:
\((x-9)^2 = x^2 – 2(x)(9) + 9^2\).
Step 3: Simplify:
\(= x^2 – 18x + 81\).
Answer: \(x^2 – 18x + 81\)

ii.  \((6-y)^2\)

Step 1: Here, \(a = 6, b = y\).
Step 2: Apply identity:
\((6-y)^2 = 6^2 – 2(6)(y) + y^2\).
Step 3: Simplify:
\(= 36 – 12y + y^2\).
Answer: \(y^2 – 12y + 36\)

iii.  \((3a-2)^2\)

Step 1: Here, \(a = 3a, b = 2\).
Step 2: Apply identity:
\((3a-2)^2 = (3a)^2 – 2(3a)(2) + 2^2\).
Step 3: Simplify:
\(= 9a^2 – 12a + 4\).
Answer: \(9a^2 – 12a + 4\)

iv.  \((8y-5z)^2\)

Step 1: Here, \(a = 8y, b = 5z\).
Step 2: Apply identity:
\((8y-5z)^2 = (8y)^2 – 2(8y)(5z) + (5z)^2\).
Step 3: Simplify:
\(= 64y^2 – 80yz + 25z^2\).
Answer: \(64y^2 – 80yz + 25z^2\)

v.  \(\left(\frac{x}{2}-\frac{y}{2}\right)^2\)

Step 1: Here, \(a = \frac{x}{2}, b = \frac{y}{2}\).
Step 2: Apply identity:
\(\left(\frac{x}{2}-\frac{y}{2}\right)^2 = \left(\frac{x}{2}\right)^2 – 2\left(\frac{x}{2}\right)\left(\frac{y}{2}\right) + \left(\frac{y}{2}\right)^2\).
Step 3: Simplify:
\(= \frac{x^2}{4} – \frac{xy}{2} + \frac{y^2}{4}\).
Answer: \(\frac{x^2}{4} – \frac{xy}{2} + \frac{y^2}{4}\)

vi.  \(\left(2a-\frac{5}{2}\right)^2\)

Step 1: Here, \(a = 2a, b = \frac{5}{2}\).
Step 2: Apply identity:
\(\left(2a-\frac{5}{2}\right)^2 = (2a)^2 – 2(2a)\left(\frac{5}{2}\right) + \left(\frac{5}{2}\right)^2\).
Step 3: Simplify:
\(= 4a^2 – 10a + \frac{25}{4}\).
Answer: \(4a^2 – 10a + \frac{25}{4}\)

vii.  \(\left(\frac{2}{a}-\frac{3}{b}\right)^2\)

Step 1: Here, \(a = \frac{2}{a}, b = \frac{3}{b}\).
Step 2: Apply identity:
\(\left(\frac{2}{a}-\frac{3}{b}\right)^2 = \left(\frac{2}{a}\right)^2 – 2\left(\frac{2}{a}\right)\left(\frac{3}{b}\right) + \left(\frac{3}{b}\right)^2\).
Step 3: Simplify:
\(= \frac{4}{a^2} – \frac{12}{ab} + \frac{9}{b^2}\).
Answer: \(\frac{4}{a^2} – \frac{12}{ab} + \frac{9}{b^2}\)

viii.  \(\left(3x-\frac{1}{3x}\right)^2\)

Step 1: Here, \(a = 3x, b = \frac{1}{3x}\).
Step 2: Apply identity:
\(\left(3x-\frac{1}{3x}\right)^2 = (3x)^2 – 2(3x)\left(\frac{1}{3x}\right) + \left(\frac{1}{3x}\right)^2\).
Step 3: Simplify:
\(= 9x^2 – 2 + \frac{1}{9x^2}\).
Answer: \(9x^2 – 2 + \frac{1}{9x^2}\)

Q3: Using special expansions, find the value of:

i. (53)²

Step 1: We know the identity (a+b)² = a² + 2ab + b²
Step 2: Write 53 as (50 + 3)
Step 3: Apply identity:
(53)² = (50+3)²
= (50)² + 2(50)(3) + (3)²
= 2500 + 300 + 9
Answer: 2809

ii. (84)²

Step 1: Identity: (a+b)² = a² + 2ab + b²
Step 2: Write 84 as (80 + 4)
Step 3: Apply identity:
(84)² = (80+4)²
= (80)² + 2(80)(4) + (4)²
= 6400 + 640 + 16
Answer: 7056

iii. (1011)²

Step 1: Identity: (a+b)² = a² + 2ab + b²
Step 2: Write 1011 as (1000 + 11)
Step 3: Apply identity:
(1011)² = (1000+11)²
= (1000)² + 2(1000)(11) + (11)²
= 1000000 + 22000 + 121
Answer: 1022121

iv. (10.9)²

Step 1: Identity: (a-b)² = a² – 2ab + b²
Step 2: Write 10.9 as (11 – 0.1)
Step 3: Apply identity:
(10.9)² = (11 – 0.1)²
= (11)² – 2(11)(0.1) + (0.1)²
= 121 – 2.2 + 0.01
Answer: 118.81

Q4: Using special expansions, find the value of:

i. \((67)^2\)

Step 1: We know the identity:
\((a – b)^2 = a^2 – 2ab + b^2\)
Step 2: Here, 67 can be written as (70 – 3).
So, \((67)^2 = (70 – 3)^2\)
Step 3: Expanding using identity:
\(= 70^2 – 2 × 70 × 3 + 3^2\)
\(= 4900 – 420 + 9\)
Answer: \((67)^2 = 4489\)

ii. \((795)^2\)

Step 1: Use the identity \((a – b)^2 = a^2 – 2ab + b^2\)
Step 2: Write 795 as (800 – 5).
So, \((795)^2 = (800 – 5)^2\)
Step 3: Expanding:
\(= 800^2 – 2 × 800 × 5 + 5^2\)
\(= 640000 – 8000 + 25\)
Answer: \((795)^2 = 632025\)

iii. \((988)^2\)

Step 1: Use identity \((a – b)^2 = a^2 – 2ab + b^2\)
Step 2: Write 988 as (1000 – 12).
So, \((988)^2 = (1000 – 12)^2\)
Step 3: Expanding:
\(= 1000^2 – 2 × 1000 × 12 + 12^2\)
\(= 1000000 – 24000 + 144\)
Answer: \((988)^2 = 976144\)

iv. \((9.2)^2\)

Step 1: Use identity \((a + b)^2 = a^2 + 2ab + b^2\)
Step 2: Write 9.2 as (9 + 0.2).
So, \((9.2)^2 = (9 + 0.2)^2\)
Step 3: Expanding:
\(= 9^2 + 2 × 9 × 0.2 + (0.2)^2\)
\(= 81 + 3.6 + 0.04\)
Answer: \((9.2)^2 = 84.64\)

Q5: If \(\left(x+\frac{1}{x}\right)=4\), find the value of:

i.  \( \left(x-\frac{1}{x}\right) \)

Step 1: Given \( x+\frac{1}{x}=4\).
Step 2: Square both sides to get \( \left(x+\frac{1}{x}\right)^2 = 4^2\).
Step 3: Expand: \( x^2 + 2 + \frac{1}{x^2} = 16\).
Step 4: So \( x^2 + \frac{1}{x^2} = 14\).
Step 5: Use \( \left(x-\frac{1}{x}\right)^2 = \left(x+\frac{1}{x}\right)^2 – 4\).
Step 6: Therefore \( \left(x-\frac{1}{x}\right)^2 = 16 – 4 = 12\).
Step 7: Hence \( x-\frac{1}{x} = \pm\sqrt{12} = \pm 2\sqrt{3}.\)
Answer: \( x-\frac{1}{x} = \pm 2\sqrt{3} \)

ii.  \( \left(x^2+\frac{1}{x^2}\right) \)

Step 1: From earlier squaring, \( \left(x+\frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 16\).
Step 2: So \( x^2 + \frac{1}{x^2} = 16 – 2 = 14.\)
Answer: \( x^2+\frac{1}{x^2} = 14 \)

iii.  \( \left(x^4+\frac{1}{x^4}\right) \)

Step 1: Square the result from part (ii): \( \left(x^2+\frac{1}{x^2}\right)^2 = 14^2\).
Step 2: Expand: \( x^4 + 2 + \frac{1}{x^4} = 196\).
Step 3: Therefore \( x^4 + \frac{1}{x^4} = 196 – 2 = 194.\)
Answer: \( x^4+\frac{1}{x^4} = 194 \)

Q6: If \(\left(z-\frac{1}{z}\right)=6\), find the values of:

i. \(\left(z+\frac{1}{z}\right)\)

Step 1: We know identity: \[ \left(z-\frac{1}{z}\right)^2 = \left(z+\frac{1}{z}\right)^2 – 4 \]Step 2: Substitute given value: \[ \left(z-\frac{1}{z}\right) = 6 \\ (6)^2 = \left(z+\frac{1}{z}\right)^2 – 4 \]Step 3: \[ 36 = \left(z+\frac{1}{z}\right)^2 – 4 \\ \left(z+\frac{1}{z}\right)^2 = 40 \]Step 4: \[ z+\frac{1}{z} = \pm \sqrt{40} = \pm 2\sqrt{10} \]Answer: \(z+\frac{1}{z} = \pm 2\sqrt{10}\)

ii. \(\left(z^2+\frac{1}{z^2}\right)\)

Step 1: We know identity: \[ \left(z+\frac{1}{z}\right)^2 = z^2 + \frac{1}{z^2} + 2 \]Step 2: From part (i): \[ \left(z+\frac{1}{z}\right)^2 = 40 \\ 40 = z^2 + \frac{1}{z^2} + 2 \]Step 3: \[ z^2 + \frac{1}{z^2} = 40 – 2 = 38 \]Answer: \(z^2+\frac{1}{z^2} = 38\)

iii. \(\left(z^4+\frac{1}{z^4}\right)\)

Step 1: We know identity: \[ \left(z^2+\frac{1}{z^2}\right)^2 = z^4 + \frac{1}{z^4} + 2 \]Step 2: From part (ii): \[ z^2 + \frac{1}{z^2} = 38 \\ (38)^2 = z^4 + \frac{1}{z^4} + 2 \]Step 3: \[ 1444 = z^4 + \frac{1}{z^4} + 2 \\ z^4 + \frac{1}{z^4} = 1442 \]Answer: \(z^4+\frac{1}{z^4} = 1442\)

Q7: If \(\left(a^2+\frac{1}{a^2}\right)=23\), find the value of \(\left(a+\frac{1}{a}\right)\).

Step 1: We know the identity:
\(\left(a+\frac{1}{a}\right)^2 = a^2 + \frac{1}{a^2} + 2\).
Step 2: Substitute the given value \(a^2+\frac{1}{a^2}=23\):
\(\left(a+\frac{1}{a}\right)^2 = 23 + 2\).
Step 3: Simplify:
\(\left(a+\frac{1}{a}\right)^2 = 25\).
Step 4: Take square root (both signs):
\(a+\frac{1}{a} = \pm \sqrt{25} = \pm 5\).
Answer: \(a+\frac{1}{a} = \pm 5\)

Q8: If \(\left(x^2+\frac{1}{x^2}\right)=102\), find the value of \(\left(x-\frac{1}{x}\right)\).

Step 1: We know the identity: \[ \left(x – \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} – 2 \]Step 2: Substitute the given value: \[ x^2 + \frac{1}{x^2} = 102 \]So, \[ \left(x – \frac{1}{x}\right)^2 = 102 – 2 \]Step 3: Simplify: \[ \left(x – \frac{1}{x}\right)^2 = 100 \]Step 4: Take square root: \[ x – \frac{1}{x} = \pm \sqrt{100} \\ x – \frac{1}{x} = \pm 10 \]Answer: \(x – \frac{1}{x} = \pm 10\)

Q9: If \(\left(2p+\frac{1}{2p}\right)=5\), find the value of \(\left(4p^2+\frac{1}{4p^2}\right)\).

i.  Find \( \left(4p^2+\frac{1}{4p^2}\right) \)

Step 1: Put \( x = 2p + \frac{1}{2p} \).
Step 2: Given \( x = 5 \).
Step 3: Square both sides: \( x^2 = \left(2p + \frac{1}{2p}\right)^2 \).
Step 4: Expand RHS: \( = (2p)^2 + 2\cdot(2p)\cdot\left(\frac{1}{2p}\right) + \left(\frac{1}{2p}\right)^2 \).
Step 5: Simplify: \( x^2 = 4p^2 + 2 + \frac{1}{4p^2} \).
Step 6: Rearrange for required expression: \( 4p^2 + \frac{1}{4p^2} = x^2 – 2 \).
Step 7: Substitute \( x = 5 \): \( 4p^2 + \frac{1}{4p^2} = 5^2 – 2 = 25 – 2 \).
Answer: \( 23 \)

Q10: If \(\left(3c-\frac{1}{3c}\right)=8\), find the value of \(\left(9c^2+\frac{1}{9c^2}\right)\).

Step 1: Let x = 3c
Then, the given equation becomes: x – 1/x = 8
Step 2: Square both sides: (x – 1/x)² = 8²
x² + 1/x² – 2 = 64
x² + 1/x² = 66
Step 3: We need (9c² + 1/(9c²)).
But x = 3c ⇒ x² = 9c².
So, 9c² + 1/(9c²) = x² + 1/x².
Answer: 66

Q11: If \(a+b=8\) and \(ab=15\), find the value of \(\left(a^2+b^2\right)\).

Step 1: We know the identity: \[ (a+b)^2 = a^2 + b^2 + 2ab \]Step 2: Rearranging for \(a^2+b^2\): \[ a^2+b^2 = (a+b)^2 – 2ab \]Step 3: Substitute the given values \(a+b=8\) and \(ab=15\): \[ a^2+b^2 = (8)^2 – 2(15) \]Step 4: Simplify: \[ a^2+b^2 = 64 – 30 \\ a^2+b^2 = 34 \]Answer: \(a^2+b^2 = 34\)

Q12: If \(a+b=11\) and \(a^2+b^2=61\), find the value of \(ab\).

Step 1: Recall the identity: \[(a+b)^2 = a^2 + b^2 + 2ab\]Step 2: Substitute given values. \[(11)^2 = 61 + 2ab\]Step 3: Simplify. \[ 121 = 61 + 2ab \]Step 4: Subtract 61 from both sides. \[ 121 – 61 = 2ab \\ 60 = 2ab \]Step 5: Divide by 2. \[ ab = \frac{60}{2} = 30 \]Answer: ab = 30

Q13: If \(a^2+b^2=13\) and \(ab=6\), find the value of \((a+b)\).

Step 1: Recall the identity: \[ (a+b)^2 = a^2 + b^2 + 2ab \]Step 2: Substitute the given values: \[ (a+b)^2 = 13 + 2(6) \\ (a+b)^2 = 13 + 12 \\ (a+b)^2 = 25 \]Step 3: Take square root on both sides: \[ a+b = \pm \sqrt{25} \\ a+b = \pm 5 \]Answer: \(a+b = \pm 5\)

Q14: If \(a+b=15\) and \(ab=56\), find the value of \(\left(a^2+b^2\right)\).

i.  Find \(a^2+b^2\)

Step 1: Recall the identity:
\((a+b)^2 = a^2 + b^2 + 2ab\).
Step 2: Rearrange to get \(a^2+b^2\):
\(a^2 + b^2 = (a+b)^2 – 2ab\).
Step 3: Substitute the given values \(a+b=15\) and \(ab=56\):
\(a^2 + b^2 = (15)^2 – 2\times 56\).
Step 4: Simplify:
\(= 225 – 112 = 113\).
Answer: \(a^2+b^2 = 113\)

Q15: If \(a-b=1\) and \(ab=12\), find the value of \(\left(a^2+b^2\right)\).

Step 1: Recall the identity: \[ (a-b)^2 = a^2 + b^2 – 2ab \]Step 2: Substitute the given values: \[ (1)^2 = a^2 + b^2 – 2(12) \]Step 3: Simplify: \[ 1 = a^2 + b^2 – 24 \]Step 4: Add 24 on both sides: \[ a^2 + b^2 = 25 \]Answer: \(a^2+b^2 = 25\)

Q16: If \(a-b=5\) and \(a^2+b^2=53\), find the value of \(ab\).

Step 1: Recall the identity \[ (a – b)^2 = a^2 + b^2 – 2ab \]Step 2: Substitute the given values. \[ (a – b)^2 = 5^2 = 25 \\ a^2 + b^2 = 53 \]So, \[ 25 = 53 – 2ab \]Step 3: Rearrange to solve for \(ab\). \[ 2ab = 53 – 25 \\ 2ab = 28 \\ ab = 14 \]Answer: ab = 14

Q17: If \(a^2+b^2=52\) and \(ab=24\), find the value of \((a-b)\).

Step 1: Recall the identity: \[ (a-b)^2 = a^2 + b^2 – 2ab \]Step 2: Substitute the given values \(a^2+b^2=52\) and \(ab=24\): \[ (a-b)^2 = 52 – 2(24) \\ (a-b)^2 = 52 – 48 \\ (a-b)^2 = 4 \]Step 3: Take the square root on both sides: \[ a-b = \pm \sqrt{4} = \pm 2 \]Answer: \(a-b = \pm 2\)

Q18: Find the value of:

i. \(36x^2 + 49y^2 + 84xy\), when x = 3, y = 6

Step 1: Observe the expression is of the form \((ax + by)^2 = a^2x^2 + b^2y^2 + 2abxy\).
Here, \(36x^2 = (6x)^2\), \(49y^2 = (7y)^2\), and \(84xy = 2 \cdot 6x \cdot 7y\).
Step 2: So, \[ 36x^2 + 49y^2 + 84xy = (6x + 7y)^2 \]Step 3: Substitute x = 3, y = 6: \[ 6(3) + 7(6) = 18 + 42 = 60 \]Step 4: Square the result: \[ (6x + 7y)^2 = 60^2 = 3600 \]Answer: 3600

ii. \(25x^2 + 16y^2 – 40xy\), when x = 6, y = 7

Step 1: Observe the expression is of the form \((ax – by)^2 = a^2x^2 + b^2y^2 – 2abxy\).
Here, \(25x^2 = (5x)^2\), \(16y^2 = (4y)^2\), and \(-40xy = -2 \cdot 5x \cdot 4y\).
Step 2: So, \[ 25x^2 + 16y^2 – 40xy = (5x – 4y)^2 \]Step 3: Substitute x = 6, y = 7: \[ 5(6) – 4(7) = 30 – 28 = 2 \]Step 4: Square the result: \[ (5x – 4y)^2 = 2^2 = 4 \]Answer: 4