Quadratic Functions

ExampleThe roots of 2 x 2 + 3 x − 1 = 0   are α and β.  Find the values of (a) ( α + 1 ) ( β + 1 ) (b) 1 α + 1 β (c) α 2 β + α β 2 (d) α β + β α [Clue: α 2 … Read more

Question 11: Diagram above shows the graphs of the curves y = x2 + x – kx + 5 and y = 2(x – 3) – 4h that intersect the x-axis at two points. Find (a) the value of k and of h, (b) the minimum value of each curve. Solution: (a) y= x 2 … Read more

Question 7: Without drawing graph or using method of differentiation, find the maximum or minimum value of the function y = 2 + 4x – 3×2. Hence, find the equation of the axis of symmetry of the graph. Solution: By completing the square for the function in the form of y = a(x + p)2+ … Read more

Question 1: (a)  Find the values of k if the equation (1 – k) x2– 2(k + 5)x + k + 4 = 0 has real and equal roots. Hence, find the roots of the equation based on the values of k obtained. (b)  Given the curve y = 5 + 4x – x2 has … Read more

Question 31 (SPM 2019):Diagram 1 shows the graph of a quadratic function$$ f(x)=\frac{p}{x^n}+q x+r $$such that p, q, r, n and u are constants.(a) State the value of n.(b) If f(x) = 0 and the product of roots is r, state the value of (i) q,(ii) p.[3 marks] Answer:(a)$$ \begin{aligned} & x^{-n}=x^2 \\ & -n=2 \\ … Read more

Question 26 (SPM 2018 – 3 marks):Faizal has a rectangular plywood with a dimension 3x metre in length and 2x metre in width. He cuts part of the plywood into a square shape with sides of x metre to make a table surface.Find the range of values of x if the remaining area of the … Read more

Question 21: Given that the quadratic equation hx2 – (h + 2)x – (h – 4) = 0 has real and distinc roots.  Find the range of values of h. Solution: The quadratic equation h x 2 −( h+2 )x−( h−4 )=0 has real and distinc roots. b 2 −4ac>0 is applied. ( −h−2 ) 2 −4( h )( −h+4 )>0 h 2 +4h+4+4 … Read more

Question 16: Find the range of values of p for which the equation 2 x 2 +5x+3−p=0 has two real distinct roots. Solution: Question 17: Find the minimum value of the function f (x) = 2×2 + 6x + 5. State the value of xthat makes f (x) a minimum value. Solution: By completing the … Read more

Question 11: Find the range of values of k for which the equation x 2 −2kx+ k 2 +5k−6=0 has no real roots. Solution: Question 12: Find the range of values of p for which the equation 5 x 2 +7x−3p=6 has no real roots. Solution: Question 13: Show that 6x−6−2k x 2 = x 2  has no real roots if k> … Read more

Question 6: Write and simplify the equation whose roots are the reciprocals of the roots of 3 x 2 +2x−1=0 , without solving the given equation. Solution: Question 7: Find the value of p if one root of x 2 +px+8=0 is the square of the other. Solution: Question 8: If one root of 2 … Read more