Question

Prove that the equation $ {x^2} + px - 1 = 0 $ has real and distinct roots for all real values of $ p $ .

Answer 1

Hint: The given equation is a quadratic equation. We are asked to prove that the given quadratic equation has real and distinct roots for any real value of $ p $ . We will use the formula to find the discriminant of the given quadratic equation and check if it satisfies the condition for real roots.
After doing this, we have to prove that the discriminant of the given quadratic equation is non-zero for any arbitrary value of $ p $ .
The discriminant of a quadratic equation $ a{x^2} + bx + c = 0 $ , where $ a \ne 0 $ is given by
 $ D = {b^2} - 4ac $
 $ D < 0 \Rightarrow $ Solution of quadratic equation is imaginary
 $ D > 0 \Rightarrow $ Solution of quadratic is real and distinct
 $ D \ne 0 \Rightarrow $ Solutions are distinct

Complete step-by-step answer:
Here, the given quadratic equation is:
 $ {x^2} + px - 1 = 0 $
 We will find the value of discriminant which is given by
 $ D = {b^2} - 4ac $ --(1)
Comparing given equation and the general form of a quadratic equation we get
 $ {x^2} + px - 1 = 0 $ --(given equation)
 $ a{x^2} + bx + c = 0 $ --(general form of quadratic equation)
S so we have the following after comparing:
 $ a = 1,b = p,c = - 1 $
Putting these in (1) we get:
 $ D = {p^2} - 4 \cdot 1 \cdot ( - 1) = {p^2} + 4 $
The value of the square of any real number is always positive or zero so $ {p^2} \geqslant 0 $ , also the sum of two positive numbers is always positive. So,
 $ D = {p^2} + 4 \geqslant 0 + 4 $
 $ \Rightarrow D \geqslant 4 $ or we can say $ D \ne 4 $
Since, the discriminant of the given quadratic equation is positive thus it yields real solutions only.
As discriminant is non-zero, so roots are distinct.
Therefore, for any real value of $ p $ the equation $ {x^2} + px - 1 = 0 $ has real and distinct roots.

Note: The proof is half solved if you only find the $ D > 0 $ condition and don’t find out that discriminant is non-zero. So make sure to show all the things clearly to complete the proof. The fact that square of any real number yields zero or positive numbers can be checked manually.