Question

Solve by quadratic formula or completion square

14. abx^2 - ( a^2 + b^2 )x + ab = 0

Ans: a / b , b / a

Standard:- 10

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Answer 1

abx² - (a² + b²)x + ab = 0

[ by quadratic formula ]

where, a' = ab ,b' = -(a² + b²) , c = ab

b'² - 4ac = [-(a² + b²)]² - 4(ab)(ab)

b'² - 4ac = a⁴ + b⁴ + 2(ab)² - 4(ab)²

b'² - 4ac = [ a⁴ + b⁴ - 2(ab)² ]

b'² - 4ac = (a² - b²)²

so, x = [ -b ± √{b² - 4ac} ]/2a

x = [ - {-(a² + b²)} ± √{(a² - b²)} ]/2(ab)

x = [ +(a² + b²) ± (a² - b²) ]/2ab

taking ( - ve).

x = [ a² + b² - a² + b² ]/2ab

x = 2b²/2ab

x = b/a

taking (+ve)

x = [ (a² + b²) + (a² - b²) ]/2ab

x = (a² + b² + a² - b²)/2ab

x = 2a²/2ab

x = a/b

so, x = a/b , x = b/a

now , by completing square root,

abx² - (a² + b²)x + ab = 0

[ multiply both side by "ab" ]

(abx)² - ab(a² + b²)x + (ab)² = 0

(abx)² - (abx)(a² + b²) + (ab)² = 0

[ (abx)² - 2(abx)(a² + b²)/2 + (a² + b²)²/4 ] - (a² + b²)²/4 + (ab)² = 0

[ abx - (a² + b²)/2 ]² - [a⁴ + b⁴ + 2(ab)²]/4 + (ab)² = 0

[ abx - (a² + b²)/2 ]² = [a⁴ + b⁴ +2(ab)² - 4(ab)² ]/4

[abx - (a² + b²)/2]² = [a⁴ + b⁴ - 2(ab)²] /4

[abx - (a² + b²)/2]² = (a² - b²)²/4

taking Square Root both side,

[abx - (a² + b²)/2] = ±(a² - b²)/2

taking (+ve).

[abx - (a² + b²)/2] = (a² - b²)/2

abx = (a² - b²)/2 + (a² + b²)/2

abx = (a² - b² + a² + b²)/2

x = 2a²/2ab

x = a/b

taking (-ve).

[abx - (a² + b²)/2 ] = -(a² - b²)/2

abx = (a² + b²)/2 - (a² - b²)/2

abx = (a² - b² - a² + b²)/2

x = 2b²/2ab

x = b/a, x = a/b

Answer 2

$\bf{hey}$

$\bf \underline{here \: is \: the \: answer}$
⬇⤵⬇⤵⬇⤵✌✌✌❕


abx²- (a²+b²)x+ab=0

(by quadratic formula)

where, a'=ab,b'=-(a²+b²),c=ab

b'²-4ac=[-(a²+b²)]²-4(ab)(ab)

b'²-4ac=a⁴+b⁴-2(ab)1

b'²-4ac=[a⁴+b⁴-2(ab)²]

b'²-4ac=(a²-b²)1

so, x= [-b ±√{b²-4ac}]/2a

x=[-{-(a²+b²)}±√{(a²-b²)/2ab

taking (-ve)

x=[a²+b²-a²+b²)/2ab

x=2b²/2ab

x=b/a

taking (+ve)

x=[(a²+b²)+(a²-b²)]/2ab

x= (a²+b²+a²-b²)/2ab

x=2a²/2ab

x=a/b

so, x= a/b

and.

x=b/a




$\underline{hope \: it \: helps}$
$\boxed{thanks}$