(a) Divide (frac{x^{2} – 4}{x^{2} + x}) by (frac{x^{2} – 4x + 4}{x + 1}).
(b) The diagram below shows the graphs of (y = ax^{2} + bx + c) and (y = mx + k) where a, b, c and m are constants. Use the graph(s) to :
(i) find the roots of the equation (ax^{2} + 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 = ax^{2} + bx + c).
Explanation
(a) (frac{x^{2} – 4}{x^{2} + x} div frac{x^{2} – 4x + 4}{x + 1})
= (frac{x^{2} – 4}{x^{2} + x} times frac{x + 1}{x^{2} – 4x + 4})
= (frac{(x – 2)(x + 2)}{x(x + 1)} times frac{x + 1}{x^{2} – 2x – 2x + 4}) (Using difference of two squares)
= (frac{(x – 2)(x + 2)}{x} times frac{1}{(x – 2)^{2}})
= (frac{(x + 2)}{x(x – 2)}.
(b) Roots of (ax^{2} +bx + c = mx + k)
For graph (y = ax^{2} + bx + c),
(x = -1) and (x = 2)
(x + 1 = 0) and (x – 2 = 0)
((x + 1)(x – 2) = 0)
(x^{2} – 2x + x – 2 = 0)
(x^{2} – x – 2 = 0 ….. (i))
(-x^{2} + x + 2 = 0 …… (ii))
For graph (y = mx + k)
(m = frac{y_{2} – y_{1}}{x_{2} – x_{1}} = frac{1 – (-1)}{1.3 – (-1.3)})
(m = frac{2}{2.6} = 0.769)
(k = 0.25)
(x^{2} – x – 2 = 0.769x + 0.25)
(x^{2} – x – 0.769x – 2 – 0.25 = 0)
(x^{2} – 1.769x – 2.25 = 0)
Using formula method, (x = frac{-b pm sqrt{b^{2} – 4ac}}{2a})
(x = frac{-(-1.769) pm sqrt{(-1.769)^{2} – 4(1)(-2.25)}}{2(1)})
(x = frac{1.769 pm sqrt{3.1294 + 9}}{2})
(x = frac{1.769 pm 3.483}{2})
(x = frac{1.769 + 3.483}{2}) or (x = frac{1.769 – 3.483}{2})
(x = frac{5.252}{2} approxeq 2.626) or (x = frac{-1.714}{2} approxeq -0.857)
(ii) The coordinates of points L, M and N are (-1, 0), (0, 2) and (2, 0) respectively.
Using coordinate for L:
(0 = a(-1^{2}) + b(-1) + c implies 0 = a – b + c ….. (1))
Using M :
(2 = a(0^{2}) + b(0) + c implies c = 2 …… (2))
Using N :
(0 = a(2^{2}) + b(2) + c implies 0 = 4a + 2b + c …. (3))
From (2), c = 2
(therefore (1) : a – b + 2 = 0 implies a – b = -2 …. (1a))
(therefore (3) : 4a + 2b + 2 = 0 implies 4a + 2b = -2 ….. (3a))
Solving equations (1a) and (3a) simultaneously, we get
(a = -1 , b = 1 , c = 2)
(therefore) Equation of the curve : (-x^{2} + x + 2 = 0)
(iii) Line of symmetry = (x = frac{-b}{2a})
(y = -x^{2} + x + 2 )
Line of symmetry = (frac{-1}{2(-1)})
(x = frac{1}{2})