If the roots of the equation (4 + m)$$x^2$$ + (m + 1)x + 1 = 0 are equal, then find the value of m.​

Correct option is A

Given: 

If the roots of the equation $$(4 + m)x^2 + (m + 1)x + 1 = 0$$ are equal , then find the value of m. 

Concept Used: 

​For a quadratic equation $$ax^2 + bx + c = 0$$, 

the condition for the roots to be equal is that the discriminant (Δ) should be zero.

The discriminant of a quadratic equation is given by:

​$$\Delta = b^2 - 4ac$$  

Solution:

The given quadratic equation is 

​$$(4 + m)x^2 + (m + 1)x + 1 = 0$$ 

​​Here, we have:

a = 4 + m

b = m + 1

c = 1

Substituting these values into the discriminant formula

​$$\Delta = (m + 1)^2 - 4(4 + m)(1)$$

​$$\Delta = (m + 1)^2 - 4(4 + m)$$

​​$$\Delta = (m^2 + 2m + 1) - 4(4 + m)$$

$$\Delta = m^2 + 2m + 1 - 16 - 4m$$

​​​$$\Delta = m^2 - 2m - 15$$

For the roots to be equal, Δ = 0:

​$$m^2 - 2m - 15 = 0 \\ \ \\ m^2 - 5m+3m - 15 = 0 \\ \ \\ m(m-5)+3( m - 5) = 0 \\ \ \\ (m+3)(m-5) = 0$$  

m = -3 or m = 5 

​Therefore, the possible values of $m$ 

 $m=\; 5$ and $m=\; -3$.

$m$