Exercise 4.2

Product = 6, Sum = -5. (Numbers are -2, -3)
x² − 2x − 3x + 6 = 0

x(x − 2) − 3(x − 2) = 0
(x − 2)(x − 3) = 0

Product = 2 × (-3) = -6, Sum = 5. (Numbers are 6, -1)
2x² + 6x − 1x − 3 = 0

2x(x + 3) − 1(x + 3) = 0
(2x − 1)(x + 3) = 0

Product = √3 × 6√3 = 18, Sum = 11. (Numbers are 9, 2)
√3x² + 9x + 2x + 6√3 = 0

√3x(x + 3√3) + 2(x + 3√3) = 0
(√3x + 2)(x + 3√3) = 0

Multiply by 8 to remove fraction:
16x² − 8x + 1 = 0

(4x − 1)² = 0

(6x)² − 2(6x)(1) + 1² = 0
It is a perfect square.

(6x − 1)² = 0

Let first number = x. Then second number = 20 − x.
x(20 − x) = 96 → 20x − x² = 96
x² − 20x + 96 = 0

Product = 96, Sum = -20. (Numbers: -12, -8)
(x − 12)(x − 8) = 0

Let integers be x and x+1.
x² + (x + 1)² = 265
x² + x² + 2x + 1 = 265 → 2x² + 2x − 264 = 0
Divide by 2: x² + x − 132 = 0

Product = -132, Sum = 1. (Numbers: 12, -11)
(x + 12)(x − 11) = 0

x = 11 or x = -12. (Since positive integer, ignore -12).

Base = x, Altitude = x − 7.
Pythagoras: x² + (x − 7)² = 17²
x² + x² − 14x + 49 = 289 → 2x² − 14x − 240 = 0
Divide by 2: x² − 7x − 120 = 0

Product = -120, Sum = -7. (Numbers: -15, 8)
(x − 15)(x + 8) = 0

x = 15 or x = -8 (Side length cannot be negative).

Articles = x. Cost per article = 2x + 2.
Total Cost = x(2x + 2) = 144
2x² + 2x − 144 = 0 → x² + x − 72 = 0

Product = -72, Sum = 1. (Numbers: 9, -8)
(x + 9)(x − 8) = 0

x = 8 or x = -9 (Articles cannot be negative).