Salinas Herrera
11/27/2023 · Primary School
e. \( 3^{x^{2}+1}=\frac{1}{3^{x}} \)
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Step-by-step Solution
To solve the equation \( 3^{x^{2}+1} = \frac{1}{3^{x}} \), we can start by rewriting the right side of the equation:
\[
\frac{1}{3^{x}} = 3^{-x}
\]
Now, we can rewrite the equation as:
\[
3^{x^{2}+1} = 3^{-x}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x^{2} + 1 = -x
\]
Now, we can rearrange this equation to form a standard quadratic equation:
\[
x^{2} + x + 1 = 0
\]
Next, we can use the quadratic formula to solve for \( x \). The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \( a = 1 \), \( b = 1 \), and \( c = 1 \). Plugging in these values, we have:
\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}
\]
\[
x = \frac{-1 \pm \sqrt{1 - 4}}{2}
\]
\[
x = \frac{-1 \pm \sqrt{-3}}{2}
\]
Since the discriminant \( 1 - 4 = -3 \) is negative, the solutions will be complex. We can express the square root of \(-3\) as \( i\sqrt{3} \):
\[
x = \frac{-1 \pm i\sqrt{3}}{2}
\]
Thus, the solutions to the equation \( 3^{x^{2}+1} = \frac{1}{3^{x}} \) are:
\[
x = \frac{-1 + i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{-1 - i\sqrt{3}}{2}
\]
Quick Answer
The solutions to the equation \( 3^{x^{2}+1} = \frac{1}{3^{x}} \) are \( x = \frac{-1 + i\sqrt{3}}{2} \) and \( x = \frac{-1 - i\sqrt{3}}{2} \).
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