ACT Math Practice Question #1211 (Medium (32)) | Test Citadel
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ACT Math Difficulty: Medium (32)

Digital ACT Math Practice Question #1211

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How many distinct real solutions does the equation $3^{2x} - 10 \cdot 3^x + 9 = 0$ have?
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Notice $x = 0 \implies 3^0 - 10(3^0) + 9 = 1 - 10 + 9 = 0$. That's one solution. Next try $x = 2 \implies 81 - 90 + 9 = 0$. That's two solutions.

Masterclass Solution & Distractor Trap Analysis

Let $u = 3^x$. Then $3^{2x} = (3^x)^2 = u^2$. The equation becomes quadratic: $u^2 - 10u + 9 = 0$. Factoring gives $(u - 9)(u - 1) = 0$, so $u = 9$ or $u = 1$. Substitute back $3^x$: $3^x = 9 \implies x = 2$, and $3^x = 1 \implies x = 0$. B...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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