Given that the quadratic equation is (2m+3)x^2+(4m-2)x-(m+1)=0.Find the value of m,if
•one of the root of the equation is negative
•one of the root of the equation is reciprocal of the other
the roots are
x = -[(2m-1)±√(6m^2+m+4)]/(2m+3)
The product of the roots is
-(m+1)/(2m+3)
If that is negative, then
(m+1)>0 and (2m+3)>0, so m > -1
(m+1)<0 and (2m+3)<0, so m < -3/2
If the roots are reciprocals, then
-(m+1) = 2m+3
3m = -4
m = -4/3
To find the value of m for the given quadratic equation, we'll use two conditions:
Condition 1: One of the roots of the equation is negative.
Condition 2: One of the roots of the equation is the reciprocal of the other.
Let's solve the equation using these conditions step-by-step:
Step 1: Set up the quadratic equation
The given quadratic equation is: (2m+3)x^2 + (4m-2)x - (m+1) = 0
Step 2: Apply the condition of one root being negative
For a quadratic equation of the form ax^2 + bx + c = 0, if one root is negative, then the product of the roots (r1 * r2) will be positive. In our equation, this means that (r1 * r2) > 0.
Using Vieta's formulas, the product of the roots is given by c / a. Therefore, we have:
(m+1) / (2m+3) > 0
Step 3: Analyze the condition
To satisfy the condition, the product (m+1) / (2m+3) must be greater than 0. This means both the numerator (m+1) and denominator (2m+3) should either be positive or negative.
Case 1: Numerator and denominator are both positive
If (m+1) > 0 and (2m+3) > 0, we have two possible inequalities:
m > -1 and m > -3/2
Taking the intersection of these two inequalities, we get m > -1.
Case 2: Numerator and denominator are both negative
If (m+1) < 0 and (2m+3) < 0, we have two possible inequalities:
m < -1 and m < -3/2
Taking the intersection of these two inequalities, we get m < -3/2. However, since we're looking for the value of m where one root is negative, this case is not valid.
Step 4: Apply the condition of one root being the reciprocal of the other
For a quadratic equation of the form ax^2 + bx + c = 0, if one root is the reciprocal of the other, then the sum of the roots (r1 + r2) will be zero. In our equation, this means that (r1 + r2) = 0.
Using Vieta's formulas, the sum of the roots is given by -b / a. Therefore, we have:
-(4m-2) / (2m+3) = 0
Step 5: Analyze the condition
To satisfy the condition, the sum (-(4m-2) / (2m+3)) must be equal to zero. This means the numerator -(4m-2) must be zero.
Solving -(4m-2) = 0, we get 4m-2 = 0, then 4m = 2, and finally, m = 1/2.
Step 6: Combine the conditions
To find the valid value of m that satisfies both conditions, we need to find the intersection of m > -1 from the first condition and m = 1/2 from the second condition.
Since m = 1/2 is the solution that satisfies both conditions, the value of m is 1/2.
Therefore, when m = 1/2, the given quadratic equation satisfies the conditions: one root is negative, and one root is the reciprocal of the other.