Question 17 of 43intermediate🔍 AnalyzeShort Answer2 marks

We have seen that (x + 3)(x + 4) = x2x^{2} + 7x + 12. Also (x + 6)(x + 7) = x2x^{2} + 13x + 42. Generalise the pattern to get an expression for (x + a)(x + b).

Correct Answer

(x + a)(x + b) = x2x^{2} + (a + b)x + ab

Exercise: Think and Reflect | Q: 1 | (Page 13)
For More Understanding

Explanation

The context demonstrates through examples like (x + 3)(x + 4) and (x + 6)(x + 7) that when multiplying two linear terms of the form (x + a) and (x + b), the resulting quadratic expression has the coefficient of x as the sum of 'a' and 'b', and the constant term as the product of 'a' and 'b'. This is explicitly shown in Example 10 of the provided text.

Solution Steps

  1. Step 1: Observe the given patterns: (x + 3)(x + 4) = x2x^{2} + 7x + 12 and (x + 6)(x + 7) = x2x^{2} + 13x + 42.

  2. Step 2: Notice that the coefficient of x is the sum of the two constants (3+4=7; 6+7=13), and the constant term is their product (34=12; 67=42).

  3. Step 3: Generalise this observation to get the expression: (x + a)(x + b) = x2x^{2} + (a + b)x + ab.

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