Math, asked by VedantSuryavanshi, 10 months ago

if x=√[(5+2√6)/(5-2√6)], show that x^2(x-10)^2=1

Answers

Answered by shrutiwaghmode100

Answer:

Basic Idea :- To deal with Square Roots use Rationalization.

Step 1 : Rationalize the Denominator in x.

Rationalizing the Denominator :- The process by which a fraction is rewritten so that the denominator contains only rational numbers.

So, Multiply and Divide by (5–√+1)(5+1)

x=5–√+15–√−1×5–√+15–√+1−−−−−−−−−−−−−−−√x=5+15−1×5+15+1

x=(5–√+1)×(5–√+1)(5–√−1)×(5–√+1)−−−−−−−−−−−−−−−−−√x=(5+1)×(5+1)(5−1)×(5+1)

x=(5–√+1)2(5–√)2−(1)2−−−−−−−−−−−√x=(5+1)2(5)2−(1)2

Use (a+b)2(a+b)2 in Numerator and (a+b)(a−b)=(a2−b2)(a+b)(a−b)=(a2−b2) in Denominator.

x=(5–√+1)2−−−−−−−−√(5–√)2−(1)2−−−−−−−−−−√x=(5+1)2(5)2−(1)2

x=5+1+25–√5−1−−−−√x=5+1+255−1

x=6+25–√2x=6+252

x=2×(3+5–√)4x=2×(3+5)4

x=3+5–√2x=3+52

Step 2 : Now to find x2x2 squaring both side.

x2=(3+5–√2)2x2=(3+52)2

x2=(3+5–√)2(2)2x2=(3+5)2(2)2

x2=9+5+65–√4x2=9+5+654

x2=14+65–√4x2=14+654

x2=7+35–√2x2=7+352

Step 3 : Now put value of (x)2x)2 and x in equation5x2−5x+15x2−5x+1

5(7+35–√2)−5(3+5–√2)+15(7+352)−5(3+52)+1

(35+155–√−15−55–√+22)(35+155−15−55+22)

(22+105–√2)(22+1052)

(2×11+55–√2)(2×11+552)

Result

5x2−5x+1=11+55–√5x2−5x+1=11+55

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