Questions Related to Algebra

x(6x² − 7y + 5) + xy(x + y) − 5xy(x + y − 5)

=>  (6x³ − 7xy + 5x + x²y + xy²) − (5x²y + 5xy² − 25xy)
=>  6x³ − 7xy + 5x + x²y + xy² − 5x²y − 5xy² + 25xy
=>  6x³ − 4x²y − 4xy² + 18xy + 5x

Final Answer: 6x³ − 4x²y − 4xy² + 18xy + 5x

Step 1: Find the sum of the two expressions
(x² − 4x + 7) + (2x² + 5x − 9)
= (x² + 2x²) + (−4x + 5x) + (7 − 9)
= 3x² + x − 2

Step 2: Let the required expression be A
We need: (3x² + x − 2) + A = 0

So,
A = −(3x² + x − 2)
= −3x² − x + 2

Final Answer:
−3x² − x + 2 must be added to get 0.

Step 1: Simplify the side expressions

First side:
y – 6x + 32 + 8 = y – 6x + 40
Second side:
x – 2y – 5z – 8 = x – 2y – 5z – 8

Step 2: Formula for perimeter of a rectangle
Perimeter = 2 × (length + breadth)
Since the wire goes twice around,
Total wire length = 2 × Perimeter = 4 × (length + breadth)

Step 3: Substitute the sides
Total wire = 4 × [(y – 6x + 40) + (x – 2y – 5z – 8)]
Simplify inside the bracket:
= 4 × [y – 6x + 40 + x – 2y – 5z – 8]
= 4 × [–5x – y – 5z + 32]

Step 4: Multiply by 4
= –20x – 4y – 20z + 128

Final Answer:
Total length of wire needed = –20x – 4y – 20z + 128

Step 1:
Double of (x − 2/x) = 2(x − 2/x)
Triple of (x + 2/x) = 3(x + 2/x)

Step 2:
Required product = [2(x − 2/x)] × [3(x + 2/x)]
= 6(x − 2/x)(x + 2/x)

Step 3:
Use the identity (a − b)(a + b) = a² − b²

Here, a = x and b = 2/x

So,
(x − 2/x)(x + 2/x) = x² − (2/x)²
= x² − 4/x²

Step 4:
Multiply by 6:
6(x² − 4/x²) = 6x² − 24/x²

Final Answer:
6x² − 24/x²

Let the required expression be x.

So,
(3a + 7b – 10) – x = (–2a – 7b + 9)
=>  x = (3a + 7b – 10) – (–2a – 7b + 9)
=>  x = 3a + 7b – 10 + 2a + 7b – 9
=>  x = (3a + 2a) + (7b + 7b) + (–10 – 9)
=>  x = 5a + 14b – 19

Final Answer:
5a + 14b – 19 should be subtracted from 3a + 7b – 10 to get –2a – 7b + 9.

Step 1: Multiply each term of (3a + 2b) by (9a² – 6ab + 4b²)

(3a + 2b)(9a² – 6ab + 4b²)
= 3a(9a² – 6ab + 4b²) + 2b(9a² – 6ab + 4b²)

Step 2: Expand
= (27a³ – 18a²b + 12ab²) + (18a²b – 12ab² + 8b³)

Step 3: Combine like terms
= 27a³ + (–18a²b + 18a²b) + (12ab² – 12ab²) + 8b³
= 27a³ + 8b³

Final Answer:
Product = 27a³ + 8b³

Step 1: Write them together
Sum = ab + (a + b) + (b + ab)

Step 2: Combine like terms
= ab + a + b + b + ab
= 2ab + a + 2b

Final Answer:
Sum = 2ab + a + 2b

Step 1: Expand the expression
7x²(3x – 9) + 3
= 7x² × 3x – 7x² × 9 + 3
= 21x³ – 63x² + 3

Step 2: For x = 4
= 21(4)³ – 63(4)² + 3
= 21(64) – 63(16) + 3
= 1344 – 1008 + 3
= 339

Value when x = 4  is 339

Step 3: For x = 6
= 21(6)³ – 63(6)² + 3
= 21(216) – 63(36) + 3
= 4536 – 2268 + 3
= 2271

Value when x = 6  is  2271

Final Answers:
Simplified expression: 21x³ – 63x² + 3
For x = 4  value is 339
For x = 6  value is  2271

Step 1:Use the identity
a² − b² = (a + b)(a − b)

Here, a = 9982 and b = 18   So,
(9982)² − (18)² = (9982 + 18)(9982 − 18)
= (10000)(9964)

Step 2: Substitute into the equation
10000x = 10000 × 9964

Step 3:Divide both sides by 10000
x = 9964

Final Answer:  x = 9964

Step 1: Use the identity
a² − b² = (a + b)(a − b)
Here, a = 38 and b = 22
(38² − 22²) = (38 + 22)(38 − 22)
= (60)(16)
= 960

Step 2: Divide by 16
(38² − 22²) / 16 = 960 / 16 = 60

Final Answer: 60

Using the identity (a + b)² – (a – b)² = 4ab

Here, a = 11pq and b = 4q

(11pq + 4q)² – (11pq – 4q)²
= 4 × (11pq) × (4q)
= 176pq²

Hence verified.

Step 1: Multiply the expressions
(x² + 2y)(x³ – 2xy + y³)
= x²(x³ – 2xy + y³) + 2y(x³ – 2xy + y³)
= (x⁵ – 2x³y + x²y³) + (2x³y – 4xy² + 2y⁴)
= x⁵ – 2x³y + 2x³y + x²y³ – 4xy² + 2y⁴
= x⁵ + x²y³ – 4xy² + 2y⁴

Step 2: Substitute x = 1 and y = –1
= (1)⁵ + (1)²(–1)³ – 4(1)(–1)² + 2(–1)⁴
= 1 + (1)(–1) – 4(1) + 2(1)
= 1 – 1 – 4 + 2
= –2

6x² – 5x + 3)(3x² + 7x – 3)
= 6x²(3x² + 7x – 3) – 5x(3x² + 7x – 3) + 3(3x² + 7x – 3)
= (18x⁴ + 42x³ – 18x²) + (–15x³ – 35x² + 15x) + (9x² + 21x – 9)

Now combine like terms:
= 18x⁴ + (42x³ – 15x³) + (–18x² – 35x² + 9x²) + (15x + 21x) – 9
= 18x⁴ + 27x³ – 44x² + 36x – 9

(3x² + 5y²)(5x² – 3y²)
=>  3x² × 5x² + 3x² × (−3y²) + 5y² × 5x² + 5y² × (−3y²)
=>  15x⁴ − 9x²y² + 25x²y² − 15y⁴
=>  15x⁴ + 16x²y² − 15y⁴

(−3x + 7) − (4x + 5)
=>  −3x + 7 − 4x − 5
=>  −7x + 2

(8x² + 7xy – 6y²) + (4x² – 3xy + 2y²) + (-4x² + xy – y²)
=>  (8x² + 4x² – 4x²) + (7xy – 3xy + xy) + (–6y² + 2y² – y²)
= > 8x² + 5xy – 5y²