Regent College Maths Department Core Mathematics 1 Quadratics
Quadratics September 011 C1 Note Quadratic functions and their graphs. The graph of y ax bx c. (i) a 0 (ii) a 0 The turning point can be determined by completing the square as we will see later. The x-coordinates of the point(s) where the curve crosses the x-axis are determined by solving ax bx c 0. The y-coordinates of the point where the curve crosses the y-axis is c. This is called the y-intercept. The discriminant of a quadratic function. b b 4ac In the quadratic formula x, the Discriminant is the name given to b 4ac. a The value of the discriminant determines how many real solutions (or roots) there are. If b 4ac 0 then ax bx c 0 has no real solutions. If b 4ac 0 then ax bx c 0 has one real solution. If b 4ac 0 then ax bx c 0 has two real solutions. Completing the square Example 1 Complete the square on x 6x 11. We can write x 6x x 3 9 and so we see that x 6x 11 ( x 3) 9 11 ( x 3). 3 is obtained by halving 6 So and x 8x 3 ( x 4) 16 3 ( x 4) 13 x x 1 ( x ) ( x ). 1 1 4 1 3 4 4 4 We can use completing the square to find the turning points of quadratics. Complete the square to find the turning point Example : Find the turning point of y x 6x 11. Completing the square on y x 6x 11 gives y x 6x 11 ( x 3). Since ( x 3) 0 we have that ( x 3) and so y. It takes that minimum value of when ( x 3) 0, that is when x 3. Hence the minimum point at 3,. Copyright www.pgmaths.co.uk - For AS, A notes and IGCSE / GCSE worksheets 11
September 011 C1 Note Solution of quadratic equations. Solution of quadratic equations by factorisation, use of the formula and completing the square There are three ways of solving quadratic equations 1. Factorising e.g. Solve x 7x 1 0. Factorising gives x 7x 1 ( x 3)( x 4) So we have ( x 3)( x 4) 0 and so x 3 or x 4 e.g. Factorise x 7x 6 : (i) Find two numbers which add to give 7 and which multiply to give 6 1, i.e. 4 and 3 (ii) Write x 7x 6 x 4x 3x 6 (iii) Factorise first two terms and second two terms to give x x 3 x x 3 x (iv) Hence write as. Completing the square. e.g. Solve x 10x 19 0 Completing the square gives x 10x 19 ( x 5) 5 19 ( x 5) 6 So we have ( x 5) 6 0. Hence we see that ( x 5) 6 and so x 5 6. Thus we have solutions of x 5 6 3. Quadratic Formula e.g. Solve x 10x 19 0 by the formula We use the fact that the solutions to the equation ax bx c 0 are b b 4ac x. a In this case a 1, b 10 and c 19 and so, using the formula, we have 10 x Solving simple cubics 3 Example: Solve x 5x 6x 0. 10 4 1 19 10 4 Each term has x in it so factorising leads to x x x Hence we have x x x 3 0. Thus the solutions are x 0, x or x 3. 5 6 0. Copyright www.pgmaths.co.uk - For AS, A notes and IGCSE / GCSE worksheets 1
1. Given that the equation kx + 1x + k = 0, where k is a positive constant, has equal roots, find the value of k. Jan 005, Q3. Given that f(x) = x 6x + 18, x 0, (a) express f(x) in the form (x a) + b, where a and b are integers. The curve C with equation y = f(x), x 0, meets the y-axis at P and has a minimum point at Q. (b) Sketch the graph of C, showing the coordinates of P and Q. The line y = 41 meets C at the point R. (c) Find the x-coordinate of R, giving your answer in the form p + q, where p and q are integers. (5) Jan 005, Q10 3. x 8x 9 (x + a) + b, where a and b are constants. (a) Find the value of a and the value of b. (b) Hence, or otherwise, show that the roots of x 8x 9 = 0 are c ± d 5, where c and d are integers to be found. 4. Factorise completely x 3 4x + 3x. May 005, Q3 Jan 006, Q1
5. x + x + 3 (x + a) + b. (a) Find the values of the constants a and b. (b) Sketch the graph of y = x + x + 3, indicating clearly the coordinates of any intersections with the coordinate axes. (c) Find the value of the discriminant of x + x + 3. Explain how the sign of the discriminant relates to your sketch in part (b). () The equation x + kx + 3 = 0, where k is a constant, has no real roots. () (d) Find the set of possible values of k, giving your answer in surd form. Jan 006, Q10 6. The equation x + px + (3p + 4) = 0, where p is a positive constant, has equal roots. (a) Find the value of p. (b) For this value of p, solve the equation x + px + (3p + 4) = 0. () May 006, Q8 7. The equation x 3x (k + 1) = 0, where k is a constant, has no real roots. Find the set of possible values of k. Jan 007, Q5 8. The equation x + kx + (k + 3) = 0, where k is a constant, has different real roots. (a) Show that k 4k 1 > 0. (b) Find the set of possible values of k. () May 007, Q7 9. Factorise completely x 3 9x. Jan 008, Q
10. Given that the equation qx + qx 1 = 0, where q is a constant, has no real roots, (a) show that q + 8q < 0. (b) Hence find the set of possible values of q. () June 008, Q8 11. The equation kx + 4x + (5 k) = 0, where k is a constant, has different real solutions for x. (a) Show that k satisfies (b) Hence find the set of possible values of k. k 5k + 4 > 0. Jan 009, Q7 1. The equation x + 3px + p = 0, where p is a non-zero constant, has equal roots. Find the value of p. 13. f(x) = x + 4kx + (3 + 11k), where k is a constant. June 009, Q6 (a) Express f(x) in the form (x + p) + q, where p and q are constants to be found in terms of k. Given that the equation f(x) = 0 has no real roots, (b) find the set of possible values of k. Given that k = 1, (c) sketch the graph of y = f(x), showing the coordinates of any point at which the graph crosses a coordinate axis. Jan 010, Q10 14. (a) Show that x + 6x + 11 can be written as where p and q are integers to be found. (x + p) + q, (b) Sketch the curve with equation y = x + 6x + 11, showing clearly any intersections with the coordinate axes. () ()
(c) Find the value of the discriminant of x + 6x + 11. () May 010, Q4 15. The equation x + (k 3)x + (3 k) = 0, where k is a constant, has two distinct real roots. (a) Show that k satisfies (b) Find the set of possible values of k. k + k 3 > 0. Jan 011, Q8 16. f(x) = x + (k + 3)x + k, where k is a real constant. (a) Find the discriminant of f(x) in terms of k. (b) Show that the discriminant of f(x) can be expressed in the form (k + a) + b, where a and b are integers to be found. () (c) Show that, for all values of k, the equation f(x) = 0 has real roots. () () 17. 4x 5 x = q (x + p), May 011, Q7 where p and q are integers. (a) Find the value of p and the value of q. (b) Calculate the discriminant of 4x 5 x. (c) Sketch the curve with equation y = 4x 5 x, showing clearly the coordinates of any points where the curve crosses the coordinate axes. May 01, Q8 18. Factorise completely x 4x 3. Jan 013, Q1 ()
19. The equation has two distinct real solutions for x. (k + 3)x + 6x + k = 5, where k is a constant, (a) Show that k satisfies k k 4 < 0. (b) Hence find the set of possible values of k. 0. 4x + 8x + 3 a(x + b) + c. (a) Find the values of the constants a, b and c. Jan 013, Q9 (b) Sketch the curve with equation y = 4x + 8x + 3, showing clearly the coordinates of any points where the curve crosses the coordinate axes. 1. Given the simultaneous equations Jan 013, Q10 where k is a non zero constant, x + y = 1 x 4ky + 5k = 0 (a) show that x + 8kx + k = 0. () Given that x + 8kx + k = 0 has equal roots, (b) find the value of k. (c) For this value of k, find the solution of the simultaneous equations.. Given that f(x) = x + 8x + 3, May 013, Q10 (a) find the value of the discriminant of f(x). (b) Express f(x) in the form p(x + q) + r where p, q and r are integers to be found. () The line y = 4x + c, where c is a constant, is a tangent to the curve with equation y = f(x). (c) Calculate the value of c. (5) May 014, Q11
3. Factorise fully 5x 9x 3. May 014_R, Q1