Theory question 107
Algebraic Expressions(a) Divide (x2 − 4)/(x2 + x) by (x2 − 4x + 4)/(x + 1).
(b) The diagram shows the graphs of y = ax2 + bx + c and y = mx + k where a, b, c, m and k are constants. Use the graph(s) to: (i) find the roots of the equation ax2 + bx + c = mx + k; (ii) determine the values of a, b and c using the coordinates of points L, M and N and hence write down the equation of the curve; (iii) determine the line of symmetry of the curve y = ax2 + bx + c.
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(a) (x − 2)(x + 2)/[x(x + 1)] × (x + 1)/(x − 2)2 = (x + 2)/[x(x − 2)].
(b)(i) The graphs meet at P and H: x ≈ −1.5 and x ≈ 1.5.
(ii) L(−1, 0), M(0, 2), N(2, 0). M gives c = 2. L: a − b + 2 = 0; N: 4a + 2b + 2 = 0. Solving: a = −1, b = 1. Curve: y = −x2 + x + 2.
(iii) The line of symmetry is halfway between the roots −1 and 2: x = 0.5.
Substitute each point's coordinates into y = ax2 + bx + c.