Binomial expansion and quadratic factorisation

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Formula

Explanation

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Worked solution

1. The area of a rectangle is given by length × width. The area is x^2 + 65x + 42 square metres, and the length is (x + 42) metres.
2. Let the width be w. Then: (x + 42) × w = x^2 + 65x + 42.
3. To find w, divide the area by the length: w = (x^2 + 65x + 42) / (x + 42).
4. Factorise the quadratic expression x^2 + 65x + 42. We need two integers that multiply to 42 and add to 65.
5. List factor pairs of 42: (1, 42), (2, 21), (3, 14), (6, 7). The pair (1, 42) adds to 43, (2, 21) adds to 23, (3, 14) adds to 17, (6, 7) adds to 13. None sum to 65.
6. Since no factor pair of 42 adds to 65, the quadratic does not factor over integers in the form (x + a)(x + b) with a and b integers.
7. Try polynomial division: divide x^2 + 65x + 42 by (x + 42).
8. Divide x^2 by x to get x. Multiply (x + 42) by x: x^2 + 42x.
9. Subtract: (x^2 + 65x + 42) - (x^2 + 42x) = 23x + 42.
10. Divide 23x by x to get 23. Multiply (x + 42) by 23: 23x + 966.
11. Subtract: (23x + 42) - (23x + 966) = -924. The remainder is -924, so division does not yield a clean factor.
12. Re-evaluate: the area is x^2 + 65x + 42, and length is (x + 42). Try factoring as (x + a)(x + b) = x^2 + (a+b)x + ab. We need a + b = 65 and ab = 42. The only integer pairs with ab = 42 are as above. None sum to 65. But if the length is (x + 42), then the width must be (x + 23), because (x + 42)(x + 23) = x^2 + 65x + 966, which does not match. Wait — check the original area: x^2 + 65x + 42. 42 is small, but 65 is large. Try (x + 1)(x + 64) = x^2 + 65x + 64. Too big. (x + 2)(x + 63) = x^2 + 65x + 126. Still too big. (x + 3)(x + 62) = x^2 + 65x + 186. All larger than 42. So no integer factor pair gives ab = 42 and a + b = 65. But the length is (x + 42), so the width must be (x + 23) to give sum 65. But (x + 42)(x + 23) = x^2 + 65x + 966, not 42. So the constant term must be 966, not 42. Therefore, the area expression must be x^2 + 65x + 966, not 42. But the question says 42. So likely a typo in the problem. But the choices are (x + 23), (x + 21), (x + 25), (x + 19). Try (x + 42)(x + 23) = x^2 + 65x + 966. 966 ≠ 42. Try (x + 42)(x + 19) = x^2 + 61x + 798. No. (x + 42)(x + 21) = x^2 + 63x + 882. No. (x + 42)(x + 25) = x^2 + 67x + 1050. No. None give 65x and 42. But 42 is too small. Unless the area is x^2 + 65x + 966, and 42 is a typo. But the choices suggest the width is (x + 23). So likely the area is x^2 + 65x + 966. Then (x + 42)(x + 23) = x^2 + 65x + 966. So width is (x + 23). Therefore, despite the constant term mismatch, the only choice that gives sum 65 when added to 42 is 23. So width is (x + 23).

Reference answer: x + 23

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