By Hugh Neill, Douglas Quadling, Julian Gilbey
Written to compare the contents of the Cambridge syllabus. natural arithmetic 1 corresponds to unit P1. It covers quadratics, capabilities, coordinate geometry, round degree, trigonometry, vectors, sequence, differentiation and integration.
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85. Try 4 substituting these numbers in the equation and see what happens. (b) Putting a=2, b=-3 and c=4, x- -(-3)±~(-3) 2 -4x2x4 - 3±-J9-32 -3±-J-23 --2x2 4 4 · But -23 does not have a square root. This means that the equation 2 2x - 3x + 4 = 0 has no roots. Try putting 2x graph of y \___ 2 - 3x + 4 in completed square form; what can you deduce about the 3x + 4 ? = 2x 2 - I \ CHAPTER 4: QUADRATICS (c) Putting a= 30, b = -11 and c = -30, x= -(-11) ± ~(-11) 2 -4 x 30 x (-30) 2 x30 = 11±-J121+3600 60 11 ±-J3721 11±61 -----60 60 So 72 = §_ or x = 60 5 x = - 50 = _1 60 6.
L you will see that in part (a) the roots of the equation jnvolved surds, in part (b) there were no roots, and in part (c) the roots were fractions. You can predict which case will arise by calculating the value of the expression under the square root sign, b 2 - 4ac, and thinking about the effect that 1 -b ± ~ b2 - 4ac . 1 h . h - d . ,, this va ue as m t e qua ratic iormu a x = . 2a roots will be integers or fractions . • If b 22 - 4ac is a perfect square, the 2 +bx+ c = 0 will have two roots .
13 y Y =-4x(x-3) 8 You can draw the graph of any function of this type which can be factorised as a(x-r)(x-s), by first noting that it cuts the x-axis at the points (r,O) and (s,O). The sign of the constant a tells you whether it 'bends upwards' (like y = x 2 ) or 'bends downwards'. 2 Fig. 14 In Figs. 3. 13 and 3. 14 different scales have been used on the two axes. If equal scales had been used the elongation in both figures would have been more obvious. 1 Sketch the graph of f(x) = 3x 2 - 1 ''" , 2x -1.
Advanced Level Mathematics: Pure Mathematics 1 by Hugh Neill, Douglas Quadling, Julian Gilbey