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algebra senior

Problem

The roots of the equation

2 x + 2 x^{- \frac{1}{2}} = 5

can be found by solving:

(A)

16 x^{2} - 92 x + 1 = 0

(B)

4 x^{2} - 25 x + 4 = 0

(C)

4 x^{2} - 17 x + 4 = 0

(D)

2 x^{2} - 21 x + 2 = 0

Solution

Make the substitution

t = x

. Then

2 t + \frac{2}{t} = 5 ⟹ 2 t^{2} - 5 t + 2 = 0

. Then

2 x - 5 x + 2 = 0 ⟹ 2 x + 2 = 5 x ⟹ (2 x + 2)^{2} = 25 x ⟹ 4 x^{2} + 8 x + 4 = 25 x ⟹ 4 x^{2} - 17 x + 4 = 0

,

4 x^{2} - 17 x + 4 = 0

Final answer

C