Question

Find the value of p if \[{x^2} - (p + 2)x + 4 = 0\] has equal roots.

Answer 1

Hint:
According to the given question, the quadratic equation is in the form of \[a{x^2} + bx + c = 0\] and it is also given that the roots are equal. So, we will firstly calculate the value of a, b and c and substitute them in the equation \[{b^2} - 4ac = 0\]. After simplifying, we can the value of p using splitting the middle term method.

Complete step by step solution:
As it is given that the above quadratic equation \[{x^2} - (p + 2)x + 4 = 0\] has equal roots.
As the equation is in the form of \[a{x^2} + bx + c = 0\] . If the equations have equal roots then \[{b^2} - 4ac = 0\] which means discriminant that is D equals zero.
In this quadratic equation \[{x^2} - (p + 2)x + 4 = 0\]. Here, \[a = 1\] , \[b = - (p + 2)\] and \[c = 4\].
Therefore, substituting all the values of a, b and c in \[{b^2} - 4ac = 0\]
We get, \[{\left( { - (p + 2)} \right)^2} - 4 \times 1 \times 4 = 0\]
Opening the square by using the identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] as \[a = p\] and \[b = 2\] and also multiply the quantities which are given we get,
\[ \Rightarrow {p^2} + 4 + 4p - 16 = 0\]
On subtracting the similar values we get,
\[ \Rightarrow {p^2} + 4p - 12 = 0\]
Now, we will calculate the roots by using splitting the middle term method.
In this method we will find out two numbers whose sum is 4 and product is \[ - 12\].
So, we are getting with the two numbers which are \[ - 2\] and \[6\]
\[ \Rightarrow {p^2} + 6p - 2p - 12 = 0\]
Taking out common in the pairs of 2 we get,
\[ \Rightarrow p\left( {p + 6} \right) - 2\left( {p + 6} \right) = 0\]
Taking 2 same factors one time we get,
\[ \Rightarrow \left( {p + 6} \right)\left( {p - 2} \right) = 0\]
Now, we will separate the above two factors to calculate the value of p.
Firstly we will take the factor\[p + 6 = 0\]
Taking 6 on the right side we get,
Therefore, \[p = - 6\]
Secondly we will take the factor \[p - 2 = 0\]
Taking 2 on the right side we get,
Therefore, \[p = 2\]

Hence, the value of \[p = 2, - 6\]

Note:
To solve these types of questions, firstly check that the quadratic equation has equal roots, distinct roots or imaginary roots respectively. If the roots are equal then \[{b^2} - 4ac = 0\] , if the roots are distinct then \[{b^2} - 4ac > 0\] and if the roots are imaginary then \[{b^2} - 4ac < 0\].