Explanation for the Algebra Quiz 04

This is the detailed explanation for the Algebra Quiz 04. Each question’s solution is broken down step-by-step to help you understand the core concepts and formulas.

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1. Factorize -4h² + 81

The correct factorization is (9 + 2h)(9 – 2h).

• This is a difference of squares. The expression can be rewritten as 81−4h2.
• Since 81=92 and 4h2=(2h)2, the expression fits the formula a2−b2=(a+b)(a−b).
• Substituting $a=9$ and $b=2h$, we get $(9+2h)(9-2h)$.

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2. Factorization of x + xy + 2y + 2y² gives

The correct factorization is (1 + y)(x + 2y).

• This expression can be factored by grouping.
• Group the first two terms and the last two terms: (x+xy)+(2y+2y2).
• Factor out the common factor from each group: $x(1+y)+2y(1+y)$.
• Now, factor out the common binomial factor $(1+y)$: $(1+y)(x+2y)$.

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3. Perimeter of rectangle is 20cm and area is 24cm² length and breadth are

The length and breadth are 6cm and 4cm.

• Let the length be $L$ and the breadth be $B$.
• Perimeter formula: $2(L+B)=20$, which simplifies to $L+B=10$.
• Area formula: $L\times B=24$.
• We need two numbers that add up to 10 and multiply to 24. These numbers are 6 and 4.
• $6+4=10$ and $6\times 4=24$.
• So, the length and breadth are 6 cm and 4 cm.

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4. Factorization of 103² – 9 yields

The correct factorization yields 10600.

• This is a difference of squares problem, a2−b2=(a−b)(a+b).
• Here, $a=103$ and $b=3$ (since 9=32).
• The factorization is $(103-3)(103+3)$.
• $(100)(106)=10600$.

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5. Expansion and simplification of 8(3h – 4) + 5(h – 2) gives

The correct simplified expression is 29h – 42.

• First, distribute the 8 and the 5:
  • $8(3h-4)=24h-32$
  • $5(h-2)=5h-10$
• Combine the results: $(24h-32)+(5h-10)$.
• Group like terms: $(24h+5h)+(-32-10)$.
• Simplify: $29h-42$.

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6. Factorization of 4x – 20 yields

he correct factorization is 4(x – 5).

• To factorize this expression, we find the greatest common factor (GCF) of the terms.
• The terms are $4x$ and $-20$.
• The GCF of 4 and 20 is 4.
• Factor out the 4 from both terms: $4(x-5)$.

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7. (a + b)² gives

The correct expansion is a² + 2ab + b².

• This is the formula for a perfect square trinomial.
• You can derive it by multiplying the binomial by itself: $(a+b)(a+b)$.
• Using the FOIL method: a×a+a×b+b×a+b×b=a2+ab+ab+b2=a2+2ab+b2.

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8. Solving following expression 3⁄(2 – a⁄6)

The correct solution is 18⁄(12 – a).

• First, find a common denominator for the terms in the denominator of the main fraction:
  • $2-a/6=12/6-a/6=(12-a)/6$.
• Now substitute this back into the original expression: $3/((12-a)/6)$.
• To divide by a fraction, multiply by its reciprocal: $3\times (6/(12-a))$.
• Simplify: 18 / (12 – a).

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9. Expansion of -8(3a + 5b) yields

The correct expansion is -24a – 40b.

• Use the distributive property to multiply -8 by each term inside the parentheses.
• $-8\times 3a=-24a$
• $-8\times 5b=-40b$
• Combining the results gives -24a – 40b.

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10. If we make a subject of given formula (p + a)⁄5 = 3p we get

The correct answer is a = 14p.

• To solve for ‘a’, first multiply both sides of the equation by 5 to eliminate the fraction:
  • $(p+a)/5\times 5=3p\times 5$
  • $p+a=15p$
• Next, subtract ‘p’ from both sides to isolate ‘a’:
  • $a=15p-p$
  • $a=\text{14p}$.