a) find, as surds, the roots of the equation
2(x+1)(x-4)-(x-2)^2 = 0
b) use algebra to solve
(x-1)(x+2)=18
c) find the values of k for which kx^2 +8x + k = 0
2(x+1)(x-4)-(x-2)^2 = 0 FOIL -->
2[x^2 -3x -4] -[x^2 -4x +4] = 0
2x^2 -6x -8 -x^2 +4x -4 = 0
2x^2 -x^2 -6x +4x -8 -4 = 0
x^2 -2x -12 = 0 Quadratic equation from here with a = 1, b = -2, c = -12
--------
(x-1)(x+2)=18 FOIL -->
x^2 +x -2 = 18 Set equal to zero [standard form for quadratics] -->
x^2 +x -20 = 0 Factor -->
(x+5)(x-4) = 0 Set each factor equal to zero and solve.
AS for part C of your qns, you are looking at the equation : kx^2 +8x + k = 0 This will require some of your basics in quadratic equations. For a simple quadratic equation, ax^2 + bx + c = 0, this will be true if b^2 - 4ac term is greater or equal to zero. So the solution to this problem follows the same concept.
kx^2 +8x + k = 0
Thus for this equation to be true, first:
(8^2) - 4(k)(k) > 0
64 - 4k^2 > 0
16 - k^2 > 0
k^2 - 16 > 0
(k - 4) (k + 4) > 0
Thus, by refering to the number line: k > 4 and K < -4
This will be the suggested solution.