Question

If a, p are the roots of the equation ( x - a ) ( x - b) + c = 0 , then the roots of the equation ( x - a) ( x - p) = c are

(a) a, b
(b) a, c
(c) b, c
(d)none of these

please give the correct answer with explanation..
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Answer 1

Answer:

Step-by-step explanation:

As we know that for the quadratic equation ax

2

+bx+c=0, roots will be equal if

D=B

2

−4AC=0

Therefore, for the equation,

a(b−c)x

2

+b(c−a)x+c(a−b)=0

A=a(b−c),B=b(c−a),C=c(a−b)

D=0

B

2

−4AC=0

(b(c−a))

2

−4(a(b−c))(c(a−b))=0

⇒ab+bc=2ac

Hence a,b and c are in HP.

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Answer 2

Answer:

if the roots of (x-a)(x-b)+c=0 are a and p, it means that if you substitute either a or p for x in the equation then it will simplify to 0

(x-a)(x-b)+c=0

(x-a)(x-b)=-c

when we substitute x with a,

(a-a)(a-b)=-c

but a-a=0

so, 0(a-b)=-c

0=-c or c=0 [1]

and with p,

(p-a)(p-b)+c=0

plugging in the value of c,

(p-a)(p-b)+0=0

or, (p-a)(p-b)=0

so p-a=0 or p-b=0

if p-a=0, p=a and there would only be one root so it's not possible

so p-b=0

p=b [2]

in the 2nd equation

(x-a)(x-p)=c

we can substitute the values of c and p from [1] and [2],

(x-a)(x-b)=0

so x-a=0 or x-b=0

therefore x=a or x=b

so the roots are a, b (option (a))