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Question

Given that x = 57 is a solution to the equation 56 x^2 + 27 = 89x - 8, what is the other value of x that will solve the equation? Express your answer as a common fraction.

Accepted Answer

First of all, let us place everything to one side:

56 x^{2} + 27 56 x^{2} - 89 x + 35 = 89 x - 8 = 0.

Now we must factor. Knowing that

x = \frac{5}{7}

is a solution to this equation, we can reason that

(7 x - 5)

must be one of the factors of

56 x^{2} - 89 x + 35

which means that

(8 x - 7)

must be the other factor, since the linear terms must multiply to

56 x^{2}

and the constant terms must multiply to

35.

We can easily verify that indeed,

56 x^{2} - 89 x + 35 = (7 x - 5) (8 x - 7),

therefore

x = \frac{7}{8}

is our answer.