Explanation for the Algebra Quiz 01

This is the detailed explanation for the Algebra Quiz 01. Each question’s solution is broken down step-by-step to help you understand the core concepts and formulas.

1. Expand and simplify (x – y)(x + y)

The simplified expression is $x^{2} - y^{2}$ .

This is a classic difference of squares formula. Using the distributive property (also known as the FOIL method), we multiply each term in the first parenthesis by each term in the second:

First: $x \times x = x^{2}$
Outer: $x \times y = x y$
Inner: $- y \times x = - x y$
Last: $- y \times y = - y^{2}$

Combining the terms, we get: $x^{2} + x y - x y - y^{2}$ . The middle terms ( $+ x y$ and $- x y$ ) cancel each other out, leaving $x^{2} - y^{2}$ .

2. Simplify 15ax^2 / 5x

The simplified expression is 3ax.

To simplify the fraction, we can simplify the numerical coefficients and the variables separately.

Numbers: $15/5 = 3$
Variables: $a$ remains as is. $x^{2} / x = x$ Combining the simplified parts, we get 3ax.

3. Simplify 5/2 ÷ 1/x

The simplified expression is 5x / 2.

To divide by a fraction, we multiply by its reciprocal. The reciprocal of $1/ x$ is $x /1$ . The expression becomes: $(5/2) \times (x /1) = (5 \times x) / (2 \times 1) = 5x / 2$ .

4. Factorize x^2 + x – 72

The factored form is (x – 8)(x + 9).

To factorize this quadratic expression, we look for two numbers that multiply to -72 and add up to 1 (the coefficient of the $x$ term).

The two numbers are 9 and -8.
$9 \times - 8 = - 72$
$9 + (- 8) = 1$ Therefore, the factored form is (x – 8)(x + 9).

5. Simplify a(c – b) – b(a – c)

The simplified expression is ac – 2ab + bc.

We distribute the terms outside the parentheses to the terms inside.

Expand $a (c - b)$ : $a c - ab$
Expand $- b (a - c)$ : $- ba - (- b c) = - ab + b c$ Combine the expanded parts: $(a c - ab) - ab + b c$ . Since $- ab$ and $- ab$ are like terms, we can combine them: $a c - 2 ab + b c$ .

6. Expand and simplify (x + y)^3

The expanded form is $x^{3} + 3 x y (x + y) + y^{3}$ .

The binomial expansion formula for a cube is $(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ . Applying this to $(x + y)^{3}$ : $x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ . The middle two terms, $3 x^{2} y$ and $3 x y^{2}$ , share a common factor of $3 x y$ . We can factor this out: $3 x^{2} y + 3 x y^{2} = 3 x y (x + y)$ . Substituting this back into the expression, we get: $x^{3} + 3 x y (x + y) + y^{3}$ .

7. Factorize -20x^2 – 9x + 20

The factored form is (5 – 4x)(4 + 5x).

To factor this quadratic, we can first factor out a -1 to make the leading coefficient positive, giving $- (20 x^{2} + 9 x - 20)$ . Now, we look for two numbers that multiply to $20 \times - 20 = - 400$ and add up to 9. The two numbers are 25 and -16. We rewrite the expression and factor by grouping: $20 x^{2} + 25 x - 16 x - 20$ $5 x (4 x + 5) - 4 (4 x + 5)$ $(5 x - 4) (4 x + 5)$ Since we factored out a negative, the final factored form is $- (5 x - 4) (4 x + 5)$ . We can distribute the negative into the first binomial to match the options: $(- 5 x + 4) (4 x + 5) = (4 - 5x)(4 + 5x)$ . Wait, this does not match the provided answer. Let’s try working directly with the negative leading coefficient. We need two binomials $(a x + b)$ and $(c x + d)$ where $a c = - 20$ and $b d = 20$ . Let’s test the option (5 – 4x)(4 + 5x): $(5 - 4 x) (4 + 5 x) = (5 \times 4) + (5 \times 5 x) + (- 4 x \times 4) + (- 4 x \times 5 x)$ $= 20 + 25 x - 16 x - 20 x^{2}$ $= - 20 x^{2} + 9 x + 20$ . This does not match the original expression which has $- 9 x$ . Let’s try (5 + 4x)(4 – 5x): $(5 + 4 x) (4 - 5 x) = (5 \times 4) + (5 \times - 5 x) + (4 x \times 4) + (4 x \times - 5 x)$ $= 20 - 25 x + 16 x - 20 x^{2}$ $= - 20 x^{2} - 9 x + 20$ . This matches the original expression. The answer is (5 + 4x)(4 – 5x).

8. (a – b)^2 =

The expanded form is $a^{2} - 2 ab + b^{2}$ .

This is a fundamental algebraic identity. We can derive it by multiplying the expression by itself: $(a - b)^{2} = (a - b) (a - b)$ Using the FOIL method:

First: $a \times a = a^{2}$
Outer: $a \times - b = - ab$
Inner: $- b \times a = - ab$
Last: $- b \times - b = + b^{2}$ Combining the terms: $a^2 – ab – ab + b^2 = \textbf{a^2 – 2ab + b^2}$.

9. Coefficient of x^2 in 4x^3 + 3x^2 – x + 1 is:

The coefficient is 3.

The coefficient of a variable in a term is the numerical factor multiplying the variable. In the given polynomial, the term containing $x^{2}$ is $3 x^{2}$ . The numerical factor is 3, so the coefficient of $x^{2}$ is 3.

10. Expand and simplify (x – 5)(x + 4)

The expanded form is $x^{2} - x - 20$ .

We use the FOIL method to expand the expression:

First: $x \times x = x^{2}$
Outer: $x \times 4 = 4 x$
Inner: $- 5 \times x = - 5 x$
Last: $- 5 \times 4 = - 20$

Combining the terms, we get: $x^{2} + 4 x - 5 x - 20$ . Combining the like terms ( $4 x - 5 x = - x$ ), we get $x^{2} - x - 20$ .

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Explanation for the Algebra Quiz 01

1. Expand and simplify (x – y)(x + y)

2. Simplify 15ax^2 / 5x

3. Simplify 5/2 ÷ 1/x

4. Factorize x^2 + x – 72

5. Simplify a(c – b) – b(a – c)

6. Expand and simplify (x + y)^3

7. Factorize -20x^2 – 9x + 20

8. (a – b)^2 =

9. Coefficient of x^2 in 4x^3 + 3x^2 – x + 1 is:

10. Expand and simplify (x – 5)(x + 4)

About the author

Muhammad Arslan

1. Expand and simplify (x – y)(x + y)

2. Simplify 15ax^2 / 5x

3. Simplify 5/2 ÷ 1/x

4. Factorize x^2 + x – 72

5. Simplify a(c – b) – b(a – c)

6. Expand and simplify (x + y)^3

7. Factorize -20x^2 – 9x + 20

8. (a – b)^2 =

9. Coefficient of x^2 in 4x^3 + 3x^2 – x + 1 is:

10. Expand and simplify (x – 5)(x + 4)

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About the author

Muhammad Arslan