Question

Find the values of p for which the quadratic equation $ p{x^2} - 6x + p = 0 $ has real and equal roots.

Answer 1

Hint: The given problem requires us to find the value of a coefficient of the given quadratic equation such that it has real and equal roots. We must know the conditions upon the discriminant of the quadratic equation for it to have real and equal roots. So, we first compare the given quadratic equation with the general form and find the discriminant. Then, we apply the specific condition upon the discriminant of the equation as the equation has real and equal roots.

Complete step-by-step answer:
In the given question, we are given the quadratic equation $ p{x^2} - 6x + p = 0 $ .
Comparing with standard form of quadratic equation $ a{x^2} + bx + c = 0 $ , we get,
 $ a = p $ , $ b = - 6 $ and $ c = p $ .
Now, we know the expression for the discriminant of a quadratic equation is $ D = {b^2} - 4ac $ .
Hence, Discriminant $ = D = {b^2} - 4ac $
Now, we know that the condition for a quadratic equation to have real and equal roots is its discriminant should be equal to zero.
So, we get, $ D = {b^2} - 4ac = 0 $
 $ \Rightarrow {\left( { - 6} \right)^2} - 4\left( p \right)\left( p \right) = 0 $
Opening the brackets and simplifying the expression, we get,
 $ \Rightarrow 36 - 4{p^2} = 0 $
Isolating p in the equation to find the value, we get,
 $ \Rightarrow 4{p^2} = 36 $
Dividing both sides of the equation by $ 4 $ , we get,
 $ \Rightarrow {p^2} = 9 $
Now, we take square roots on both sides of the equation to find the value of p. Hence, we get,
 $ \Rightarrow p = \pm 3 $
Therefore, the values of p for which the equation $ p{x^2} - 6x + p = 0 $ has real and equal roots are $ 3 $ and $ - 3 $ .
So, the correct answer is “ $ \pm 3 $ ”.

Note: Quadratic equations are the polynomial equations with degree of the variable or unknown as two. We must know how to calculate the expression for the discriminant of a quadratic equation in order to solve such a problem as it provides us an idea of the nature of roots of the equation. We also must know the general form of the quadratic equation.