Question 54

If $$30x^2 - 15x + 1 = 0$$, then what is the value of $$25x^2 + (36x^2)^{-1}$$?

Solution

$$30x^2 - 15x + 1 = 0$$

Dividing by x,

$$30x - 15 + \frac{1}{x} = 0$$

$$5x - 15/6 + \frac{1}{6x} = 0$$

$$5x + \frac{1}{6x} = 5/2$$

taking square both side,

$$(5x + \frac{1}{6x})^2 = 25/4$$

$$25x^2 + \frac{1}{36x^2} + 2 \times 5x \times \frac{1}{6x} = 25/4$$

$$25x^2 + \frac{1}{36x^2} = 25/4 - 5/3$$

$$25x^2 + \frac{1}{36x^2} = \frac{55}{12} $$

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