(a) Simplify (frac{x + 2}{x – 2} – frac{x + 3}{x – 1})
(b) The graph of the equation (y = Ax^{2} + Bx + C) passes through the point (0, 0), (1, 4) and (2, 10). Find the :
(i) value of C ; (ii) values of A and B ; (iii) co-ordinates of the other point where the graph cuts the x- axis.
Explanation
(a) (frac{x + 2}{x – 2} – frac{x + 3}{x – 1})
= (frac{(x – 1)(x + 2) – (x – 2)(x + 3)}{(x – 2)(x – 1)})
= (frac{x^{2} + 2x – x – 2 – (x^{2} + 3x – 2x – 6)}{(x – 2)(x – 1)})
= (frac{x^{2} – x^{2} + x – x – 2 + 6}{(x – 2)(x – 1)})
= (frac{4}{(x – 2)(x – 1)})
(b)
(1, 4) and (2, 10) are supposed roots of the equation
When x = 1, equation becomes (A(1)^{2} + B(1) = 4)
When x = 2, (A(2^{2}) + B(2) = 10)
(implies A + B = 4 …… (1))
(4A + 2B = 10 ……. (2))
Solving for A and B, we have
A = 1 and B = 3.
(iii) (y = x^{2} + 3x)
(y = x(x + 3))
x = 0 or x = 3.