Cambridge University Entrance Algebra Exam

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[anantkulkarni]

It can be, by observation two times x-1 is a factor of 4 degree polynomial then by synthetic division we get third quadratic polynomial or equation by quadratic formula we get irrational roots

[YAWTon]

Straightforward solution: square both sides and simplify to arrive at the quartic equation
x^4-7x^3+12x^2-7x+1=0. The quartic equation has an obvious solution x=1: divide by (x-1):
x^3-6x^2+6x-1=0: this has the obvious solution x=-1: divide by (x-1) again:
x^2-5x+1=1: use formula for quadratic equation to obtain two additional solutions x=(5+√21)/2 and x=(5-√21)/2. Check that the three values of x satisfy the original equation (check needed because squaring in the first step might have introduced spurious solutions).

[destruidor]

You should pay more attention when you make substition….such x by a…as a teach, i give you a zero mark for such error.

[walterwen]

Cambridge University Entrance Algebra Exam: x = √[27/(x – 4) – 20/(x – 3)]; x =?
x ≠ 3, 4; x = √{[27(x – 3) – 20(x – 4)]/[(x – 3)(x – 4)]} = √{(7x – 1)/[(x – 3)(x – 4)]}
x² = (7x – 1)/[(x – 3)(x – 4)], x²(x – 3)(x – 4) = 7x – 1, x²(x² – 7x + 12) – 7x + 1 = 0
(x⁴ – 7x³ + 10x²) + 2x² – 7x + 1 = 0, (x² – 2x)(x² – 5x) + (2x² – 7x) + 1 = 0
(x² – 2x + 1)(x² – 5x + 1) = [(x – 1)²](x² – 5x + 1) = 0; x – 1 = 0 or x² – 5x + 1 = 0
x = 1, Double root or x = (5 ± √21)/2
Answer check:
x = 1: x = √[27/(x – 4) – 20/(x – 3)] = √[27/(– 3) – 20/(– 2)] = 1; Confirmed
x = (5 ± √21)/2, x² = [(5 ± √21)/2]² = (46 ± 10√21)/4 = (23 ± 5√21)/2
x – 4 = (5 ± √21)/2 – 4 = (– 3 ± √21)/2, x – 3 = (– 1 ± √21)/2
x² = 27/(x – 4) – 20/(x – 3) = 54/(– 3 ± √21) – 40/(– 1 ± √21)
= [54(3 ± √21)]/(21 – 9) – [40(1 ± √21)]/(21 – 1)
= (27 ± 9√21)/2 – (2 ± 2√21) = (23 ± 5√21)/2; Confirmed
Final answer:
x = 1, Double root; x = (5 + √21)/2 or x = (5 – √21)/2

[nasrullahhusnan]

x²(x-4)(x-3)=7x-1
x⁴-7x³+12x²-7x+1=0
(x-1)(x³+ax²+bx-1)=0
x⁴+(a-1)x³+(b-a)x²-(1+b)x+1=0
a-1=-7 --> a=-6
b-a=12 --> b=12+(-6)=6
Thus (x-1)(x³-6x²+6x-1)=0
(x-1)²(x²-5x+1)=0
x=1
x²-5x+1=0 --> x=½[5±sqrt(21)]