If f(x)= $4 x^{2} + 3 x - 7$ and α is a common root of the equation $x^{2} - 3 x + 2 = 0$ and $x^{2} + 2 x - 3 = 0$ then find the value of f(α)

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Detailed Solution

Firstly we will take the equation,

x^{2} - 3 x + 2 = 0

Now, we will calculate the roots by using the middle term method.
In this method we will find out two numbers whose sum is −3 and product is 2.
So, we are getting with the two numbers which are −2 and −1
⇒

x^{2} - x - 2 x + 2 = 0

Taking out common in the pairs of 2 we get,
⇒x(x−1)−2(x−1)=0
Taking 2 same factors one time we get,
⇒(x−1)(x−2)=0
Now, we will separate the above two factors to calculate the value of x.
Firstly we will take the factor x−1=0
Taking 1 on the right side we get,
Therefore, x=1
Secondly we will take the factor x−2=0
Taking 2 on the right side we get,
Therefore, x=2
Hence, the value of x=1,2
Then we will take the equation,

x^{2} + 2 x - 3 = 0

Now, we will calculate the roots by splitting the middle term method.
In this method we will find out two numbers whose sum is 2 and product is −3.
So, we are getting with the two numbers which are −1 and 3
⇒

x^{2} - x + 3 x - 3 = 0

Taking out common in the pairs of 2 we get,
⇒x(x−1)+3(x−1)=0
Taking 2 same factors one time we get,
⇒(x−1)(x+3)=0
Now, we will separate the above two factors to calculate the value of p.
Firstly we will take the factor x−1=0
Taking 1 on the right side we get,
Therefore, x=1
Secondly we will take the factor x+3=0
Taking 3 on the right side we get,
Therefore, x= −3
Hence, the value of x= 1, −3
As it is given in the question that

x^{2} - 3 x + 2 = 0

and

x^{2} + 2 x - 3 = 0

and there is a common root that is α.
But, it is clear from the calculated roots that α = 1(As 1 is common root in both the equations)
So, we will calculate f(α)= f(1) by substituting 1 in the given equation that is f(x)=