Solving for m with Equal Roots

The problem requires finding the value(s) of m for the quadratic equation $(4 + m)x^2 + (m + 1)x + 1 = 0$ when its roots are equal.

Condition for Equal Roots

A quadratic equation $ax^2 + bx + c = 0$ has equal roots if its discriminant, $D$, is equal to zero. The discriminant is calculated as:

$D = b^2 - 4ac$

For the given equation, we identify the coefficients:

• $a = 4 + m$
• $b = m + 1$
• $c = 1$

Applying the Discriminant Condition

Setting the discriminant to zero:

$D = (m + 1)^2 - 4(4 + m)(1) = 0$

Calculating Values of m

Now, we solve the equation for m:

1. Expand the terms: $(m^2 + 2m + 1) - (16 + 4m) = 0$
2. Simplify the equation: $m^2 + 2m + 1 - 16 - 4m = 0$
3. Combine like terms: $m^2 - 2m - 15 = 0$
4. Factor the quadratic equation for m: $(m - 5)(m + 3) = 0$
5. Determine the possible values for m: $m - 5 = 0$ or $m + 3 = 0$
6. Therefore, the values are: $m = 5$ or $m = -3$.

Conclusion

The values of m for which the equation $(4 + m)x^2 + (m + 1)x + 1 = 0$ has equal roots are $5$ and $-3$.