Question

x= √√5+1/√5-1 find 5x²-5x-1​

Answer 1

Answer:

If x=√ [( √5+1) ÷ (√5-1)], then what is the value of 5x^2-5x+1?

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

Step 1 : Rationalize the Denominator in x.

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

So, Multiply and Divide by (5–√+1)

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

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

x=(5–√+1)2(5–√)2−(1)2−−−−−−−−−−−√

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

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

x=5+1+25–√5−1−−−−√

x=6+25–√2

x=2×(3+5–√)4

x=3+5–√2

Step 2 : Now to find x2 squaring both side.

x2=(3+5–√2)2

x2=(3+5–√)2(2)2

x2=9+5+65–√4

x2=14+65–√4

x2=7+35–√2

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

5(7+35–√2)−5(3+5–√2)+1

(35+155–√−15−55–√+22)

(22+105–√2)

(2×11+55–√2)

Result

5x2−5x+1=11+55–√