Question 1

If (a + b)² = 81 and ab = 18, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Substitute (a + b)² = 81 and ab = 18 into the identity: 81 = a² + 2(18) + b². Step 3: Simplify: 81 = a² + 36 + b². Step 4: Therefore, a² + b² = 81 - 36 = 45.

Question 2

Simplify: (x + 5)² - (x - 5)²

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Step 2: Expand (x + 5)² = x² + 10x + 25. Step 3: Expand (x - 5)² = x² - 10x + 25. Step 4: Subtract: (x² + 10x + 25) - (x² - 10x + 25) = x² + 10x + 25 - x² + 10x - 25 = 20x.

Question 3

If a - b = 7 and ab = 12, find a² + b².

Accepted Answer

Step 1: Use the identity (a - b)² = a² - 2ab + b². Step 2: Substitute a - b = 7 and ab = 12: (7)² = a² - 2(12) + b². Step 3: Simplify: 49 = a² - 24 + b². Step 4: Therefore, a² + b² = 49 + 24 = 73.

Question 4

Expand: (2x + 3y)(2x - 3y)

Accepted Answer

Step 1: Use the difference of squares identity (a + b)(a - b) = a² - b². Step 2: Here, a = 2x and b = 3y. Step 3: Apply: (2x)² - (3y)² = 4x² - 9y².

Question 5

If a² + b² = 34 and ab = 15, find the value of (a + b)² + (a - b)².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b² = 34 + 2(15) = 64. Step 2: Use the identity (a - b)² = a² - 2ab + b² = 34 - 2(15) = 4. Step 3: Add them: 64 + 4 = 68.

Question 6

If x + y = 12 and x² + y² = 90, then find the value of x³ + y³.

Accepted Answer

Step 1: From x + y = 12, square both sides: (x + y)² = 144, so x² + 2xy + y² = 144. Step 2: Since x² + y² = 90, we have 90 + 2xy = 144, giving xy = 27. Step 3: Use the identity x³ + y³ = (x + y)³ - 3xy(x + y) = 12³ - 3(27)(12) = 1728 - 972 = 756.

Question 7

Simplify: (x + 5)² - (x - 5)²

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Step 2: Expand (x + 5)² = x² + 10x + 25. Step 3: Expand (x - 5)² = x² - 10x + 25. Step 4: Subtract: (x² + 10x + 25) - (x² - 10x + 25) = x² + 10x + 25 - x² + 10x - 25 = 20x.

Question 8

Simplify: (a + b + c)² - (a + b - c)² using algebraic identities.

Accepted Answer

Step 1: Use the identity (x + y)² - (x - y)² = 4xy. Step 2: Let x = (a + b) and y = c. Step 3: Then (a + b + c)² - (a + b - c)² = 4(a + b)(c). Step 4: Expand: 4(a + b)(c) = 4ac + 4bc.

Question 9

If (a + b)² = 64 and ab = 12, find the value of a² + b².

Accepted Answer

Step 1: Expand (a + b)² using the identity (a + b)² = a² + 2ab + b². Step 2: Substitute the given values: 64 = a² + 2(12) + b². Step 3: Simplify: 64 = a² + 24 + b². Step 4: Therefore, a² + b² = 64 - 24 = 40.

Question 10

Simplify: (x + 5)² - (x - 5)² using algebraic identities.

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here, a = x and b = 5. Step 3: Apply the formula: (x + 5)² - (x - 5)² = 4(x)(5) = 20x.

Question 11

Simplify: (p + q)³ - (p - q)³

Accepted Answer

Step 1: Use the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a - b)³ = a³ - 3a²b + 3ab² - b³. Step 2: Expand (p + q)³ = p³ + 3p²q + 3pq² + q³. Step 3: Expand (p - q)³ = p³ - 3p²q + 3pq² - q³. Step 4: Subtract: (p³ + 3p²q + 3pq² + q³) - (p³ - 3p²q + 3pq² - q³) = p³ + 3p²q + 3pq² + q³ - p³ + 3p²q - 3pq² + q³ = 6p²q + 2q³ = 2q(3p² + q²).

Question 12

If x + 1/x = 5, find x² + 1/x².

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2(x)(1/x) + 1/x². Step 2: Simplify: (x + 1/x)² = x² + 2 + 1/x². Step 3: Substitute x + 1/x = 5: (5)² = x² + 2 + 1/x². Step 4: Simplify: 25 = x² + 2 + 1/x². Step 5: Therefore, x² + 1/x² = 25 - 2 = 23.

Question 13

Expand: (2m + 3n)(2m - 3n)

Accepted Answer

Step 1: Recognize this as the difference of squares pattern (a + b)(a - b) = a² - b². Step 2: Here a = 2m and b = 3n. Step 3: Apply the formula: (2m)² - (3n)² = 4m² - 9n².

Question 14

Expand (2x + 3y)² and find the coefficient of xy.

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Here, a = 2x and b = 3y. Step 3: Expand: (2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y². Step 4: The coefficient of xy is 12.

Question 15

If x + 1/x = 5, find the value of x² + 1/x².

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2(x)(1/x) + 1/x². Step 2: Simplify: (x + 1/x)² = x² + 2 + 1/x². Step 3: Rearrange: x² + 1/x² = (x + 1/x)² - 2. Step 4: Substitute x + 1/x = 5: x² + 1/x² = 5² - 2 = 25 - 2 = 23.

Question 16

If (a + b)² = 81 and ab = 18, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Substitute (a + b)² = 81 and ab = 18 into the identity: 81 = a² + 2(18) + b². Step 3: Simplify: 81 = a² + 36 + b². Step 4: Therefore, a² + b² = 81 - 36 = 45.

Question 17

If a - b = 7 and ab = 18, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a - b)² = a² - 2ab + b². Step 2: Rearrange to get a² + b² = (a - b)² + 2ab. Step 3: Substitute the given values: a² + b² = 7² + 2(18) = 49 + 36 = 85.

Question 18

If p - q = 7 and pq = 12, find the value of p² + q².

Accepted Answer

Step 1: Use the identity (p - q)² = p² - 2pq + q². Step 2: Rearrange to get p² + q² = (p - q)² + 2pq. Step 3: Substitute (p - q) = 7 and pq = 12: p² + q² = 7² + 2(12) = 49 + 24 = 73.

Question 19

If p + q = 12 and p² + q² = 90, find the value of (p + q)³ - 3pq(p + q).

Accepted Answer

Step 1: From p + q = 12, we get (p + q)² = 144. Step 2: Expand: p² + 2pq + q² = 144, so 90 + 2pq = 144, giving pq = 27. Step 3: Use the identity (p + q)³ - 3pq(p + q) = p³ + q³. Step 4: Calculate: (12)³ - 3(27)(12) = 1728 - 972 = 756.

Question 20

If x + 1/x = 5, then find the value of x² + 1/x².

Accepted Answer

Step 1: We know that (x + 1/x)² = x² + 2 + 1/x². Step 2: Substitute x + 1/x = 5: (5)² = x² + 2 + 1/x². Step 3: 25 = x² + 2 + 1/x². Step 4: x² + 1/x² = 25 - 2 = 23.

Question 21

Simplify: (a + b)³ - (a - b)³.

Accepted Answer

Step 1: Use the identity (p)³ - (q)³ = (p - q)(p² + pq + q²). Step 2: Here p = (a + b) and q = (a - b). Step 3: (a + b) - (a - b) = 2b. Step 4: (a + b)² = a² + 2ab + b². Step 5: (a - b)² = a² - 2ab + b². Step 6: (a + b)² + (a + b)(a - b) + (a - b)² = (a² + 2ab + b²) + (a² - b²) + (a² - 2ab + b²) = 3a² + b². Step 7: Final answer: 2b(3a² + b²) = 6a²b + 2b³.

Question 22

If a² + b² = 34 and ab = 15, find the value of (a + b)².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Rearrange: (a + b)² = (a² + b²) + 2ab. Step 3: Substitute a² + b² = 34 and ab = 15. Step 4: (a + b)² = 34 + 2(15) = 34 + 30 = 64.

Question 23

Simplify: (x + 2)² - (x - 2)².

Accepted Answer

Step 1: Use the identity p² - q² = (p + q)(p - q). Step 2: Here p = (x + 2) and q = (x - 2). Step 3: (x + 2) + (x - 2) = 2x. Step 4: (x + 2) - (x - 2) = 4. Step 5: (x + 2)² - (x - 2)² = (2x)(4) = 8x.

Question 24

If x - 1/x = 3, then find the value of x³ - 1/x³.

Accepted Answer

Step 1: Use the identity a³ - b³ = (a - b)(a² + ab + b²). Step 2: We need x³ - 1/x³ = (x - 1/x)(x² + 1 + 1/x²). Step 3: First find x² + 1/x². Step 4: (x - 1/x)² = x² - 2 + 1/x², so 9 = x² - 2 + 1/x². Step 5: x² + 1/x² = 11. Step 6: x³ - 1/x³ = (3)(11 + 1) = (3)(12) = 36.

Question 25

If a + b + c = 9 and ab + bc + ca = 26, find the value of a² + b² + c².

Accepted Answer

Step 1: Use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Rearrange: a² + b² + c² = (a + b + c)² - 2(ab + bc + ca). Step 3: Substitute a + b + c = 9 and ab + bc + ca = 26. Step 4: a² + b² + c² = (9)² - 2(26) = 81 - 52 = 29.

Question 26

Simplify: (m + n + p)² - (m + n - p)² using algebraic identities.

Accepted Answer

Step 1: Use the identity A² - B² = (A + B)(A - B). Step 2: Let A = (m + n + p) and B = (m + n - p). Step 3: A + B = (m + n + p) + (m + n - p) = 2(m + n). Step 4: A - B = (m + n + p) - (m + n - p) = 2p. Step 5: Therefore, (A + B)(A - B) = 2(m + n) × 2p = 4p(m + n).

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