1965 AHSME Problems/Problem 1

Problem

The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is:

$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 3 \qquad \textbf{(E) }\ \text{more than 4}$

Solution 1

Solution by e_power_pi_times_i

Take the logarithm with a base of $2$ to both sides, resulting in the equation $2x^2-7x+5 = 0$ . Factoring results in $(2x-5)(x-1) = 0$ , so there are $\boxed{\textbf{(C) } 2}$ real solutions.

Solution 2

Notice that $a^0=1, a>0$ . So $2^0=1$ . So $2x^2-7x+5=0$ . Evaluating the discriminant, we see that it is equal to $7^2-4*2*5=9$ . So this means that the equation has two real solutions. Therefore, select $\fbox{C}$ .

~hastapasta

1965 AHSME (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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1965 AHSME Problems/Problem 1

Contents

Problem

Solution 1

Solution 2

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