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Hi.

To start this question, take f(x) and g(x) and substitute them into the equation f(x)+tg(x)=0. Then you'll have to multiply t into g(x), and rearrange the resultant quadratic equation into the form ax^2+bx+c (multiply out tg(x) and factorise the x^2 and x terms). We then have a quadratic, and given the fact that there is only one real root to the quadratic, we can use the discriminant and say that b^2-4ac = 0. You'll then have another quadratic with the in it, which you can solve and continue with the question. 

Feel free to ask if you need more help.

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