Solve for $$ x $$ : $$ 3^{2 x+2}-10.3^{x}+1=0 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
3^{2x + 2} - 10 \cdot 3^x + 1 = 0
$$

**Step 1: Simplify the Equation**

First, express $$3^{2x + 2}$$ in terms of $$3^x$$:

$$
3^{2x + 2} = 3^{2x} \cdot 3^2 = 9 \cdot (3^x)^2
$$

Let $$y = 3^x$$. The equation becomes:

$$
9y^2 - 10y + 1 = 0
$$

**Step 2: Solve the Quadratic Equation**

Solve for $$y$$ using the quadratic formula:

$$
y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 1}}{2 \cdot 9} = \frac{10 \pm \sqrt{100 - 36}}{18} = \frac{10 \pm 8}{18}
$$

This gives two solutions for $$y$$:

$$
y_1 = \frac{18}{18} = 1 \quad \text{and} \quad y_2 = \frac{2}{18} = \frac{1}{9}
$$

**Step 3: Solve for $$x$$**

Recall that $$y = 3^x$$:

1. For $$y_1 = 1$$:

$$
3^x = 1 \implies x = 0
$$

2. For $$y_2 = \frac{1}{9}$$:

$$
3^x = \frac{1}{9} = 3^{-2} \implies x = -2
$$

**Final Answer:**

$$
\boxed{\left\{\, -2,\;0\,\right\}}
$$
The solutions are $$ x = -2 $$ and $$ x = 0 $$.

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