(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.