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The quadratic formula is a powerful tool to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this, we can find the roots of any quadratic equation. The symbols \( \pm \) signify that there are typically two solutions, one involving the plus and the other involving the minus.
This is why quadratics are often called "polynomials of degree two," since they can yield two solutions.In the given problem, the quadratic equation is \( 3a^2 - 28a + 9 = 0 \).
By substituting \( a = 3 \), \( b = -28 \), and \( c = 9 \) into the quadratic formula, we find the values of \( a \).
This step is essential as it transforms the equation into a recognizable and solvable format, uncovering the roots.

Variable substitution is a strategic method to simplify equations, especially when dealing with complex expressions involving exponentials or fractions.
In our exercise, we substitute \( a = 3^x \). This effectively turns the initially difficult equation\[3(3^x) + 9(3^{-x}) = 28\]into a more manageable form:\[3a + \frac{9}{a} = 28\]This tactic helps eliminate the complexities involved with the exponents and allows the equation to be expressed in terms of a single variable, \( a \), bridging the gap to the next step.
It's a clever approach, making subsequent steps—like transforming to a quadratic equation—much easier to handle. Once the quadratic solutions are found, we can substitute back to determine \( x \).

The discriminant, \( b^2 - 4ac \), is a crucial part of the quadratic formula.
It tells us not only the nature but also the number of solutions a quadratic equation has.- If the discriminant is positive, there are two distinct real solutions.- If it's zero, there is exactly one real solution.- If it's negative, the solutions are complex and not real.In our problem, we calculated the discriminant for \( 3a^2 - 28a + 9 = 0 \) as:\[(-28)^2 - 4 \cdot 3 \cdot 9 = 676\]A positive discriminant (676) indicates there are two real and distinct solutions for \( a \).
Knowing this provides clarity on how many solutions to expect and their nature.

In the final step, the relationship between the quadratic solution (\( a \)) and the original exponential form (\( 3^x \)) must be explored.Exponentiation refers to the operation of raising numbers to powers; here, it simplifies understanding equations like \( 3^x = 9 \).
Calculating roots means finding a number that, when raised to a certain power, gives another number.The solutions of the quadratic equation \( a = 9 \) and \( a = \frac{1}{3} \) translate to exponential forms as:- \( 3^x = 9 \) implies \( x = 2 \) since \( 9 \) is \( 3^2 \).- \( 3^x = \frac{1}{3} \) implies \( x = -1 \) because \( \frac{1}{3} = 3^{-1} \).Understanding these inverse operations helps navigate back from the solution of \( a \) to the sought-after values of \( x \).
This demonstrates the seamless transition from solving a quadratic equation back to the context of exponentiation.