Question

Find ‘m’ so that the roots of the equation $ (4 + m){x^2} + (m + 1)x + 1 = 0 $ may be equal.
a)5,-3
b)5,3
c)4,2
d)1,-5

Answer 1

Hint: The given equation is a quadratic equation and we are going to use the quadratic formula to find the root. We are asked to find the value of ‘m’ such that the two roots are equal. So we will equate the two roots that we will obtain from this given equation after applying the quadratic formula. This will give us the value of ‘m’ and we will check which option matches our obtained answer.
Formula used:
1)The general form of a quadratic equation is $ a{x^2} + bx + c = 0 $ where ‘a’ is non-zero coefficient. The roots of this equation is given by a formula called the quadratic formula. The quadratic formula is:
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
2) $ {(a + b)^2} = {a^2} + 2ab + {b^2} $

Complete step-by-step answer:
The given equation is:
 $ (4 + m){x^2} + (m + 1)x + 1 = 0 $
The given equation is identical to the general form of a quadratic equation
 $ a{x^2} + bx + c = 0 $
Here we have,
 $ a = (4 + m),b = (m + 1),c = 1 $
Now the quadratic formula is $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
Putting the respective values of a,b and c we get:
 $ x = \dfrac{{ - (m + 1) \pm \sqrt {{{(m + 1)}^2} - 4 \times (4 + m) \times 1} }}{{2 \times (4 + m)}} $
 $ \Rightarrow x = \dfrac{{ - m - 1 \pm \sqrt {{m^2} + 2m + 1 - 16 - 4m} }}{{8 + 2m}} $ [Applying formula (2) i.e. $ {(a + b)^2} = {a^2} + 2ab + {b^2} $ ]
 $ \Rightarrow x = \dfrac{{ - m - 1 \pm \sqrt {{m^2} - 2m - 15} }}{{8 + 2m}} $
Let $ {x_1} = \dfrac{{ - m - 1 + \sqrt {{m^2} - 2m - 15} }}{{8 + 2m}} $ and $ {x_2} = \dfrac{{ - m - 1 - \sqrt {{m^2} - 2m - 15} }}{{8 + 2m}} $
According to the question we want to have $ {x_1} = {x_2} $
 $ \Rightarrow \dfrac{{ - m - 1 + \sqrt {{m^2} - 2m - 15} }}{{8 + 2m}} = \dfrac{{ - m - 1 - \sqrt {{m^2} - 2m - 15} }}{{8 + 2m}} $
As the denominators are same we can cancel then and write,
 $ - m - 1 + \sqrt {{m^2} - 2m - 15} = - m - 1 - \sqrt {{m^2} - 2m - 15} $
On cancellation of similar terms on both the sides of the equation we gwt,
 $ \sqrt {{m^2} - 2m - 15} = - \sqrt {{m^2} - 2m - 15} $
 $ \Rightarrow 2\sqrt {{m^2} - 2m - 15} = 0 $
 $ \Rightarrow {m^2} - 5m + 3m - 15 = 0 $
 $ \Rightarrow m(m - 5) + 3(m - 5) = 0 $
 $ \Rightarrow (m + 3)(m - 5) = 0 $
Now we have two possibilities that are: $ (m + 3) = 0 $ or $ (m - 5) = 0 $
So $ m = - 3 $ or $ m = 5 $
As option ‘a’ matches our obtained answer we will choose it as the correct option.
So, the correct answer is “Option A”.

Note: As options are given to us, we can solve the question alternatively by just putting the values of ‘m’ given in the options in the given equation containing ‘m’ and obtaining the roots using quadratic formula. If the obtained roots come the same for a particular option then we can guarantee that the option is correct.