Question

Determine all real solutions of the equation

(x^2-p)^(1/2)+2(x^2-1)^(1/2)= x
where p is a real number.​

Answer 1

Step-by-step explanation:

The discriminant of a quadratic equation

ax

2

+bx+c=0 is given by b

2

−4ac

a=2,b=2(p+1) and c=p

[2(p+1)]

2

−4(2p)⇒4(p+1)

2

−8p

⇒4[(p+1)

2

−2p]⇒4[(p

2

+2p+1)−2p]

⇒4(p

2

+1)

For any real value of p,4(p

2

+1) will always be positive as p

2

cannot be negative for real p.

Hence, the discriminant b

2

−4ac will always be positive.

When the discriminant is greater than 0 or is positive, then the roots of a quadratic equation will be real.

Answer 2

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