Sure, let's solve the given equation step-by-step.
The original equation is:
[tex]\[ 3 \cdot 9^x + 3^{x+2} = 1 + 3^{x-1} \][/tex]
We introduce the substitution [tex]\( p = 3^x \)[/tex]. Using this substitution, we can express all terms involving [tex]\(3^x\)[/tex] in terms of [tex]\(p\)[/tex].
1. Rewrite [tex]\(9^x\)[/tex]:
[tex]\[ 9^x = (3^2)^x = (3^x)^2 = p^2 \][/tex]
2. Rewrite [tex]\(3^{x+2}\)[/tex]:
[tex]\[ 3^{x+2} = 3^x \cdot 3^2 = p \cdot 9 = 9p \][/tex]
3. Rewrite [tex]\(3^{x-1}\)[/tex]:
[tex]\[ 3^{x-1} = \frac{3^x}{3} = \frac{p}{3} \][/tex]
Now substitute these expressions back into the original equation:
[tex]\[ 3 \cdot p^2 + 9p = 1 + \frac{p}{3} \][/tex]
To eliminate the fraction, multiply every term by 3:
[tex]\[ 3 \cdot 3 \cdot p^2 + 3 \cdot 9p = 3 \cdot 1 + 3 \cdot \frac{p}{3} \][/tex]
[tex]\[ 9p^2 + 27p = 3 + p \][/tex]
Next, we bring all terms to one side of the equation to form a quadratic equation:
[tex]\[ 9p^2 + 27p - p - 3 = 0 \][/tex]
Combine like terms:
[tex]\[ 9p^2 + 26p - 3 = 0 \][/tex]
Therefore, the equation [tex]\(3 \times 9^x + 3^{x+2} = 1 + 3^{x-1}\)[/tex] can indeed be rewritten as:
[tex]\[ 9p^2 + 26p - 3 = 0 \][/tex]
where [tex]\( p = 3^x \)[/tex].
