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We Distribute, Yet Things Multiply - Mind the Mistake, Mend the Mistake

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Distributive Property states that a(b+c)=ab+aca(b + c) = ab + ac. A common mistake is to multiply the first term and forget the second.

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When squaring a binomial like (a+b)2(a + b)^2, the result is a2+2ab+b2a^2 + 2ab + b^2. Never assume (a+b)2=a2+b2(a + b)^2 = a^2 + b^2; this is the most common error in Grade 8 algebra.

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When a negative sign is outside the parentheses, it changes the sign of every term inside when distributing: −a(b−c)=−ab+ac-a(b - c) = -ab + ac.

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Multiplying two binomials (x+a)(x+b)(x + a)(x + b) follows the FOIL method or horizontal distribution: x(x+b)+a(x+b)=x2+(a+b)x+abx(x + b) + a(x + b) = x^2 + (a + b)x + ab.

📐Formulae

a(b+c)=ab+aca(b + c) = ab + ac

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab

💡Examples

Problem 1:

Expand 4(3x−5)4(3x - 5). Identify the common mistake and provide the correct solution.

Solution:

4(3x−5)=12x−204(3x - 5) = 12x - 20

Explanation:

Mistake: Writing 12x−512x - 5 by forgetting to multiply the 44 with the second term. Correction: Use a(b−c)=ab−aca(b - c) = ab - ac to get 4(3x)−4(5)=12x−204(3x) - 4(5) = 12x - 20.

Problem 2:

Evaluate (x+6)2(x + 6)^2 using algebraic identities.

Solution:

(x+6)2=x2+12x+36(x + 6)^2 = x^2 + 12x + 36

Explanation:

Mistake: Writing x2+36x^2 + 36. Correction: You must include the middle term 2ab2ab. Here a=xa = x and b=6b = 6, so 2ab=2(x)(6)=12x2ab = 2(x)(6) = 12x.

Problem 3:

Multiply (x+2)(x + 2) and (x+3)(x + 3) using the vertical method.

Solution:

x+2×x+33x+6x2+2xx2+5x+6\begin{array}{r} x + 2 \\ \times \quad x + 3 \\ \hline 3x + 6 \\ x^2 + 2x \\ \hline x^2 + 5x + 6 \end{array}

Explanation:

To avoid mistakes in distribution, align like terms vertically. First, multiply 33 by (x+2)(x + 2) to get 3x+63x + 6. Then multiply xx by (x+2)(x + 2) to get x2+2xx^2 + 2x. Finally, add the partial products.

Problem 4:

Simplify −3(x−4)-3(x - 4).

Solution:

−3x+12-3x + 12

Explanation:

Mistake: Writing −3x−12-3x - 12. Correction: When distributing a negative number, the signs change. Negative times negative equals positive: (−3)×(−4)=+12(-3) \times (-4) = +12.

Mind the Mistake, Mend the Mistake Class 8 Notes & Examples