Advanced Level Mathematics: Pure Mathematics 1 by Hugh Neill, Douglas Quadling, Julian Gilbey PDF

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X= 2x2 4 4 Sometimes you will be expected to leave the roots in surd form. 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.

2 The equation kx 2 - 2x - 7 = 0 has two real roots. What can you deduce about the value of the constant -k? The discriminant is (-2) 2 - 4 x k x (-7) = 4 + 28k. As the equation has two real roots, the value of the discriminant is positive, so 4 + 28k > 0, and k > - ~ . 3 The equation 3x 2 + 2x + k = 0 has a repeated root. Find the value of k . j.. - 4 x 3 x k = 0. This Notice how, in the above examples, there is no need to solve the quadratic equation. You can find all you need to know from the discriminant.

2 Express 2x 2 + lOx + 7 in completed square form. Start by taking out the factor 2 from the terms which involve x: 2x 2 + lOx + 7 = 2( x 2 + 5x) + 7. Dealing with the term inside the bracket, x so 2 + 5x = ( x + ~y - 2J , 2x 2 +10x+7=2(x 2 +5x)+7=2{(x+~Y- 2;}+1 2) 2 - = 2(x + 2 25 + 7 = 2(x + 2 2 2 2) -112· It's worth checking your result mentally at this stage. 3. 3 Express 3 - 4x - 2x 2 in completed square form. Start by taking out the factor - 2 from the terms which involve x: 3 - 4x - 2x 2 = 3 - 2( x 2 + 2x).

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