14. Find the value of '$m$' if roots are equal of the
a. $(1+m) x^2-2(1+3 m) x+(1+8 m)=0$

Find the discriminant D = b 2 − 4 a c of the quadratic equation.
Set the discriminant equal to zero, D = 0 , for equal roots.
Solve the equation 4 m 2 − 12 m = 0 for m .
The solutions are m = 0 and m = 3 , thus m = 0 , 3 ​ .

Explanation

Problem Analysis We are given a quadratic equation ( 1 + m ) x 2 − 2 ( 1 + 3 m ) x + ( 1 + 8 m ) = 0 and we need to find the values of m for which the roots of the equation are equal.

Equal Roots Condition For a quadratic equation of the form a x 2 + b x + c = 0 , the roots are equal if the discriminant, D = b 2 − 4 a c , is equal to zero.

Identifying Coefficients and Discriminant In our equation, we have a = ( 1 + m ) , b = − 2 ( 1 + 3 m ) , and c = ( 1 + 8 m ) . Let's calculate the discriminant:


D = b 2 − 4 a c = [ − 2 ( 1 + 3 m ) ] 2 − 4 ( 1 + m ) ( 1 + 8 m )

Expanding the Discriminant Now, we expand and simplify the discriminant:

D = 4 ( 1 + 6 m + 9 m 2 ) − 4 ( 1 + 9 m + 8 m 2 ) D = 4 + 24 m + 36 m 2 − 4 − 36 m − 32 m 2 D = 4 m 2 − 12 m

Setting Discriminant to Zero For equal roots, we set the discriminant to zero:

4 m 2 − 12 m = 0 4 m ( m − 3 ) = 0

Solving for m Solving for m , we get two possible values:

m = 0 or m = 3

Final Answer Therefore, the values of m for which the given quadratic equation has equal roots are m = 0 and m = 3 .

Examples
Consider designing a suspension system for a vehicle. The system's performance can be modeled by a quadratic equation where the roots represent the damping characteristics. If you want the system to have critically damped behavior (equal roots), you would solve for the parameter 'm' (representing a component's property) to ensure the discriminant is zero. This ensures optimal stability and ride comfort without oscillations.

Answered by GinnyAnswer | 2025-07-03

The values of m for which the given quadratic equation has equal roots are m = 0 and m = 3 . This is found by setting the discriminant of the equation to zero. The discriminant is calculated and set to zero to find the potential values for m .
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Answered by Anonymous | 2025-07-04