Question

please solve this and verify ​

Answer 1

Solution

Given :-

• x = (√5 - √3)/(√5 + √3)
• y = (√5 + √3)/(√5 - √3)

To Prove:-

• x² + y² - 6xy = 56

Explanation

Rationaliza denominator of x .

==> x = (√5 - √3)/(√5 + √3)

==> x = (√5 - √3)(√5 - √3)/(√5 + √3)(√5 - √3)

Using Formula

★(a + b)² = a² + b² + 2ab

★ (a - b)(a + b) = (a² - b²)

==> x = [(√5)² + (√3)² - 2×(√5) × (√3) ]/[ (√5)² - (√3)²]

==> x = [ 5 + 3 - 2√15]/( 5 - 3)

==> x = (8 - 2√15)/(2)

==> x = 2(4 - √15)/2

==> x = (4 - √15)

Squaring both side

==> x² = (4 - √15)²

==> x² = 4² + (√15)² - 2×4 × √15)

==> x² = 16 + 15 - 8√15

==> x² = 31 - 8√15

Now, Rationalize denominator of y

==> y = (√5 + √3)/(√5 - √3)

==>y = (√5 + √3)(√5 + √3)/(√5 - √3)(√5 + √3)

==> y = [(√5)² + (√3)² + 2 × √5 × √3]/[(√5)² - (√3)²]

==> y = (5 + 3 + 2√15)/(5 - 3)

==> y = (8 + 2√15)/2

==> y = 2(4 + √15)/2

==> y = (4 + √15)

Squaring both side

==> y² = (4 + √15)²

==> y² = 16 + 15 + 2 × 4× √15

==> y² = 31 + 8√15

Now, Calculate ( x² + y² - 6xy)

Keep value of x² , y² & xy

= ( 31 - 8√15) +(31 + 8√15) - 6 (4 - √15)(4 + √15)

= 62 - 6(16 - 15)

= 62 - 6

= 56

= R.H.S.

That's Proved.

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