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The roots of the equation

2^{2 x + 1} - 9 \times 2^{x} + 4 = 0

are $a$ and $b$ .

Find $a b$ .

Hint

This can be written

2 \times {(2^{x})}^{2} - 9 \times 2^{x} + 4 = 0

Hint

Let $u = 2^{x}$ and solve

2 u^{2} - 9 u + 4 = 0

Solution

The equation can be written as

2 \times {(2^{x})}^{2} - 9 \times 2^{x} + 4 = 0

Now let $u = 2^{x}$ to get

\begin{aligned} 2 u^{2} - 9 u + 4 & = 0 \\ (2 u - 1) (u - 4) & = 0 \end{aligned}

The solutions are

\begin{aligned} u & = \frac{1}{2} \\ 2^{x} & = \frac{1}{2} \\ = 2^{- 1} \\ x & = - 1 \end{aligned}

and

\begin{aligned} u & = 4 \\ 2^{x} & = 4 \\ x & = 2 \end{aligned}