Question

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Answer 1

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Answer 2

⠀⠀|| ☆.QUESTION.☆ ||

If the quadratic equation px² - 2√5 px + 15 = 0 has two equal roots, then find the value o p .

⠀|| ☆.ANSWER.☆ ||

GIVEN EQUATION:-

px² - 2√5 px + 15 = 0 .....(1)

A/C to question, equation (1) have two roots equal

Let, a and b are two roots ,

Then,

• a = b .....(2)

Now,

Sum Of Roots = [ -(Coefficient o x )/(Coefficient of x² ]

↪ ( a + b ) = [ -(2√5 p )/(p)]

↪ ( a + a ) = [ -2√5 ] . [by (2) ]

↪ 2a = [ -2√5 ]

↪ a = (-2√5)/2

↪a = -√5. .........(3)

And,

Product Of Roots = [ (Constant part )/(Coefficient o x² )]

↪ab = [ (15)/p]

↪(a × a) = [ 15/p ]. [by (2) ]

↪ a² = [15/p ]

↪ (-√5)² = [ 15 /p ]. [ by (3) ]

↪ p = [ 15/5 ]

↪ p = 3

Thus:-

• Value of p = 3 .

__________________

ANSWER VERIFICATION :-

Keep Value of p in equation (1) .

↪ 3x² - 6√5 x + 15 =0

Sum of Roots = [ -(6√5)/(3)]

↪ ( a+ b ) = -2√5 ......(4)

Product of Roots = [ 15/3]

↪ ab = 5 .....(5)

We know,

(a - b ) = √{(a+b)² - 4ab }

Keep value by (4) and (5)

↪ (a - b) = √{(-2√5)²-4×5}

↪ (a-b) = √{20-20}

↪ (a-b) = 0 .......(6)

Sum of equation (4) and (6) ,

↪ 2a = -2√5

↪ a = -2√5/2

↪ a = -√5 .............(7)

Keep value in (6)

↪ -√5 - b = 0

↪ b = -√5. ............(8)

By equation (7) and (8)

We can, say that Equation (1) have two roots equal

i.e.

• a = b

Proved .