Question

If the roots of the equation x2 - ax + b = 0 differ by 4, then which of the given options is true?​

Answer 1

the roots of the equation x2 - ax + b = 0 differ by 4, then which of the given options is true?

Answer 2

Given:

If the roots of the equation x² - ax + b = 0 differ by 4.

To Find:

Value of α and β.

Solution:

Let α and β be two roots of the given quadratic equation x² - ax + b = 0.

So,

as we know,

for quadric equation mx² + nx + c = 0

Sum of roots = -n/m

product of roots = c/m

therefore, for x² - ax + b = 0

α + β = -(-a)/1 = a

α.β = b

and

α - β = 4  { as per given information }

we know that,

⇒ (α + β)² = (α - β)² + 4α.β

⇒ a² = 4² + 4.b

⇒ a² = 16 + 4b

α + β = a ----1

α - β = 4 ------2

add 2 and 1

2α = 4 + a

α = 2+ a/2 = (4 + a)/2

put α = 2 + a/2 in

2 + a/2 + β = a

2 + β = a - a/2

β = a/2 -2

β = (a - 4 )/ 2

hence, α =  (4 + a)/2 and β = (a - 4 )/ 2.