Given that x=3 is a solution to f(x)= 2x^3 - 8x^2 + 7x - 3 = 0, solve f(x)=0 completely.

As x=3 is a solution, (x - 3) is a factor of f(x) = 2x3- 8x2+ 7x - 3 = 0 ie. 2x3- 8x2+ 7x - 3 = (x - 3)(Ax2 + Bx + C) where A,B,C are constants to be found. Expanding and then grouping terms by power, we obtain 2x3- 8x2+ 7x - 3 = Ax3+ (B - 3A)x2 + (C - 3B)x - 3C. Observe that A is the only coefficient of the x3 component, and therefore A = 2. By substituting A = 2 into the equation B - 3A = - 8, we have B = 3A - 8 = 3(2) - 8 = - 2 ie. B = -2. Observe that C is the only non-x coefficient and therefore C = 1. Therefore 2x3- 8x2+ 7x - 3 = (x - 3)(2x2 - 2x + 1)Now using the quadratic formula x = (-b +/- sqrt(b2 - 4ac))/2a, factorise 2x2 - 2x + 1. Giving x = (1 +/- i)/2. Therefore we have that x = 3, (1 + i)/2 and (1 - i)/2 are the three solutions of f(x)=0.

AS
Answered by Adrian S. Maths tutor

6882 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

What is a derivative and how do we calculate it from first principles?


For the curve y = 2x^2+4x+5, find the co-ordinates of the stationary point and determine whether it is a minimum or maximum point.


Integrate sin^2(x)


Solve (3x+6)/4 - (2x-6)/5 = (x+7)/8.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning