Quadratic Functions

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 9: Given that the quadratic function f(x) = 2×2 – px + p has a minimum value of –18 at x = 1. (a) Find the values of p and q. (b) With the value of p and q found in (a), find the values of x, where graph f(x) cuts the x-axis. (c) … 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 5: Given 3t and (t – 7) are the roots of the quadratic equation 4×2 – 4x + m = 0 where m is a constant. (a)  Find the values of t and m. (b)  Hence, form the quadratic equation with roots 4t and 2t + 6. Solutions: (a) Given 3t and (t – … Read more

Question 3: If α and β are the roots of the quadratic equation 3×2 + 2x– 5 = 0, form the quadratic equations that have the following roots. (a)  2 α  and  2 β (b)  ( α + 2 β )  and  ( β + 2 α ) Solution: 3×2 + 2x – 5 = … 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 20: Find the range of values of k if the quadratic equation 3(x2 – kx – 1) = k – k2 has two real and distinc roots. Solution: 3( x 2 −kx−1 )=k− k 2 3 x 2 −3kx−3−k+ k 2 =0 3 x 2 −3kx+ k 2 −k−3=0 a=3,b=−3k,c= k 2 −k−3 In cases of two real and distinc roots, … Read more

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 square for f (x) in the form of f (x) = a(x + p)2 + q to find the minimum value of f … Read more

Question 13: Show that 6x−6−2k x 2 = x 2  has no real roots if k> 1 4 . Solution: Question 14: The quadratic equation x 2 +px+q=0 has roots –2 and 6. Find (a) the value of p and of q, (b) the range of values of r for which the equation x 2 +px+q=r has no real roots. Solution: Question … Read more

Question 10: The quadratic equation x 2 −4x−1=2p(x−5) , where p is a constant, has two equal roots. Calculate the possible values of p. Solution: 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 … Read more