Question

Without using a calculator, find the positive root of the equation (5 − 2√2)x 2 − (4 + 2√2)x − 2 = 0 giving your answer in the form a + b√2, where a and b are integers.

Answer 1

Answer:

(5–2√2)x^2-(4+2√2)x-2=0

x=[(4+2√2)±√{(4+2√2)^2–4(5–2√2)(-2)]/2(5–2√2)

=[(4+2√2)±√(16+16√2+8+40–16√2)]/(10–4√2)

=[(4+2√2)±√64]/(10–4√2)

=(4+2√2±8)/(10–4√2)

when x=(4+2√2+8)/(10–4√2)

x=(12+2√2)/(10–4√2)

=(6+√2)/(5–2√2)

={(6+√2)(5+2√2)}/(25–8)

=(30+5√2+12√2+4)/17

=(34+17√2)/17

=(2+√2)

when x=(4+2√2–8)/(10–4√2)

x=(2√2–4)/(10–4√2)

=(√2–2)/(5–2√2)

=(√2–2)(5+2√2)/(25–8)

=(5√2–10+4–4√2)/17

=(√2–6)/17 as 6 is > √2 ,so (√2–6) is - ve and rejected .

So positive root =2+√2 ans