The difference between the two roots of the equation
x²+ax+b=0 is same as the difference between the two roots of the equation x²+cx+d=0.
Find (a²-c²)/(b-d)
Explain in detail.
Answers
Answer:
Required numeric value of ( a^2 - c^2 ) / ( b - d ) is 4.
Step-by-step explanation:
Given equations are x^2 + ax + b = 0 and x^2 + cx + d = 0. It is given that the difference between the roots of the equations is equal.
By Quadratic Formula, root of a quadratic equation : { - b ± √( b^2 - 4ac ) } / 2a
On comparing the given equation{ x^2 + ax + b = 0 } with a₁x^2 + b₁x + c = 0 , we get a₁ = 1 , b₁ = a , c = b .
Thus,
= > Difference between the roots : -
= > [ - b + √( b^2 - 4ac ) ] / 2a - [ - b - √( b^2 - 4ac ) ] / 2a
= > [ - a + √{ a^2 - 4( 1 )( b ) } ] / 2( 1 ) - [ [ - a - √{ a^2 - 4( 1 )( b ) } ] / 2( 1 ) ]
= > [ - a + √( a^2 - 4b ) + a + √( a^2 - 4b ) ] / 2
= > √( a^2 - 4b )
Similarly, on comparing the given equation{ x^2 + ax + b = 0 } with a₁x^2 + b₁x + c = 0 , we get a₁ = 1 , b₁ = a , c = b .
Thus,
= > Difference between the roots : -
= > [ - b + √( b^2 - 4ac ) ] / 2a - [ - b - √( b^2 - 4ac ) ] / 2a
= > [ - c + √{ c^2 - 4( 1 )( d ) } ] / 2( 1 ) - [ [ - c - √{ c^2 - 4( 1 )( d ) } ] / 2( 1 ) ]
= > [ - c + √( c^2 - 4d ) + c + √( c^2 - 4d ) ] / 2
= > √( c^2 - 4d )
Since, both the differences are equal : -
= > √( a^2 - 4b ) = √( c^2 - 4d )
= > a^2 - 4b = c^2 - 4d
= > a^2 - c^2 = 4b - 4d
= > a^2 - c^2 = 4( b - d )
= > ( a^2 - c^2 ) / ( b - d ) = 4
Hence the required numeric value of ( a^2 - c^2 ) / ( b - d ) is 4.
Answer:
the required numeric value of ( a^2 - c^2 ) / ( b - d ) is 4.