Find the solutions of the following equations which have common roots: $$4x^{4} + 12x^{3} - x^{2} - 15x = 0, 6x^{4} + 13x^{3} - 4x^{2} - 15x = 0$$.
Let $$f(x) = 4x^{4} +12x^{3} - x^{2} - 15x \quad \text{and} \quad g(x) = 6x^{4} + 13x^{3} - 4x^2 - 15x$$
H.C.F. of $$f(x)$$ and $$g(x)$$ is $$x(x-1)(2x-3)$$
Dividing $$f(x)$$ and $$g(x)$$ by $$x(x-1)(2x-3)$$, we get
$$f(x) = x(x-1)(2x-3)(2x+5) \quad \text{and} \quad g(x) = x(x-1)(2x-3)(3x+5)$$
$$\therefore\,\,$$ the roots of $$f(x) = 0$$ are $$0, 1, -\dfrac{3}{2}, -\dfrac{5}{2}$$
and the roots of $$g(x) = 0$$ are $$0, 1, -\dfrac{3}{2}, -\dfrac{5}{3}$$
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