For what value of 'm', the equation $(3m+1)x^{2}+2(m+1)x+m=0$ have equal root ?
- 1 or $-\frac{1}{2}$
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2 or 4
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4
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3
For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant D = b^2 - 4ac must be zero. Substituting the given coefficients, 4(m+1)^2 - 4(3m+1)(m) = 0 simplifies to 4(m^2 + 2m + 1) - 4(3m^2 + m) = 0, which results in -8m^2 + 4m + 4 = 0, or 2m^2 - m - 1 = 0. Factoring gives (2m+1)(m-1) = 0, so m = 1 or -1/2.
For the equation (3m + 1)x^2 + 2(m + 1)x + m = 0 to have equal roots, its discriminant must be zero, so b^2 - 4ac = 0. Substituting the coefficients gives [2(m + 1)]^2 - 4(3m + 1)(m) = 0, which simplifies to 4m^2 + 8m + 4 - 12m^2 - 4m = 0. Solving the resulting quadratic equation, 8m^2 - 4m - 4 = 0 or 2m^2 - m - 1 = 0, provides the values m = 1 or m = -1/2.