If 5x² - 8y - 20x + 2xy = (x + a) (bx + 2y), then the value of (a - b) is :

Concept

To factor a quadratic expression in two variables as a product of two linear binomials, expand the proposed product and match the coefficient of each distinct term — x2, xy, x, and y — on both sides. Each unknown is fixed by equating the coefficient of the term it appears in. This is the method of undetermined coefficients.

Application

Expand the right side (x + a)(bx + 2y):

1. (x + a)(bx + 2y) = bx2 + 2xy + abx + 2ay.

2. Match the x2 coefficient: b = 5.

3. Match the x coefficient: ab = -20. With b = 5, a = -20 / 5 = -4.

4. Cross-check the y coefficient: 2a should equal -8, and 2(-4) = -8. Consistent.

5. The xy coefficient is 2 on both sides, which matches automatically.

6. Therefore a = -4 and b = 5, so a - b = -4 - 5 = -9.

Cross-check

Substitute a = -4, b = 5 back: (x - 4)(5x + 2y) = 5x2 + 2xy - 20x - 8y, which is exactly 5x2 - 8y - 20x + 2xy. The expansion reproduces the original expression, confirming a - b = -9.