Solve The Equation For \[$x\$\]:$\[ 3^{2x} - 12 \times 3^x + 27 = 0 \\]

Feb 28, 2025 by ADMIN 72 views

**Solving the Equation for x: A Step-by-Step Guide**

Introduction

In this article, we will delve into solving a quadratic equation that involves exponential terms. The given equation is $3^{2x} - 12 \times 3^x + 27 = 0$ . This equation can be solved using various methods, including factoring, the quadratic formula, and substitution. We will explore each of these methods and provide a step-by-step solution to the equation.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$ , where $a = 1$ , $b = -12$ , and $c = 27$ . However, the equation involves exponential terms, making it a bit more complex than a standard quadratic equation.

Method 1: Factoring

One way to solve this equation is by factoring. We can start by recognizing that the equation can be written as a perfect square trinomial:

$3^{2x} - 12 \times 3^x + 27 = (3^x - 3)^2 = 0$

This allows us to factor the equation as:

$(3^x - 3)^2 = 0$

Solving for x

To solve for x, we can take the square root of both sides of the equation:

$3^x - 3 = 0$

Adding 3 to both sides gives us:

$3^x = 3$

Taking the logarithm base 3 of both sides gives us:

$x = 1$

Method 2: Quadratic Formula

Another way to solve this equation is by using the quadratic formula. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$ , $b = -12$ , and $c = 27$ . Plugging these values into the quadratic formula gives us:

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(27)}}{2(1)}$

Simplifying the expression gives us:

$x = \frac{12 \pm \sqrt{144 - 108}}{2}$

$x = \frac{12 \pm \sqrt{36}}{2}$

$x = \frac{12 \pm 6}{2}$

This gives us two possible solutions for x:

$x = \frac{12 + 6}{2} = 9$

$x = \frac{12 - 6}{2} = 3$

Method 3: Substitution

Another way to solve this equation is by using substitution. We can let $y = 3^x$ and rewrite the equation as:

$y^2 - 12y + 27 = 0$

This is a quadratic equation in y, which can be factored as:

$(y - 3)^2 = 0$

Solving for y gives us:

$y - 3 = 0$

$y = 3$

Substituting back $y = 3^x$ gives us:

$3^x = 3$

Taking the logarithm base 3 of both sides gives us:

$x = 1$

Conclusion

In this article, we have explored three different methods for solving the equation $3^{2x} - 12 \times 3^x + 27 = 0$ . We have shown that the equation can be solved using factoring, the quadratic formula, and substitution. Each of these methods provides a different solution for x, and we have demonstrated how to use each method to find the solution.

Final Answer

The final answer to the equation is $x = 1$ . This solution can be obtained using any of the three methods discussed in this article.

Additional Resources

For more information on solving quadratic equations, including those with exponential terms, we recommend the following resources:

Introduction

In our previous article, we explored three different methods for solving the equation $3^{2x} - 12 \times 3^x + 27 = 0$ . We demonstrated how to use factoring, the quadratic formula, and substitution to find the solution for x. In this article, we will provide a Q&A guide to help you better understand the solution and address any questions you may have.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = 1$ . This solution can be obtained using any of the three methods discussed in our previous article.

Q: Why did we use factoring to solve the equation?

A: We used factoring to solve the equation because it allowed us to recognize that the equation can be written as a perfect square trinomial. This made it easier to factor the equation and solve for x.

Q: Can we use the quadratic formula to solve the equation?

A: Yes, we can use the quadratic formula to solve the equation. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$ , $b = -12$ , and $c = 27$ . Plugging these values into the quadratic formula gives us:

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(27)}}{2(1)}$

Simplifying the expression gives us:

$x = \frac{12 \pm \sqrt{144 - 108}}{2}$

$x = \frac{12 \pm \sqrt{36}}{2}$

$x = \frac{12 \pm 6}{2}$

This gives us two possible solutions for x:

$x = \frac{12 + 6}{2} = 9$

$x = \frac{12 - 6}{2} = 3$

Q: Can we use substitution to solve the equation?

A: Yes, we can use substitution to solve the equation. We can let $y = 3^x$ and rewrite the equation as:

$y^2 - 12y + 27 = 0$

This is a quadratic equation in y, which can be factored as:

$(y - 3)^2 = 0$

Solving for y gives us:

$y - 3 = 0$

$y = 3$

Substituting back $y = 3^x$ gives us:

$3^x = 3$

Taking the logarithm base 3 of both sides gives us:

$x = 1$

Q: What if I get a different solution using a different method?

A: If you get a different solution using a different method, it's possible that the solution is not valid. In this case, you should recheck your work and make sure that you have not made any mistakes. If you are still unsure, you can try using a different method to solve the equation.

Q: Can I use this method to solve other equations?

A: Yes, you can use this method to solve other equations that involve exponential terms. However, you should make sure that the equation can be written in the form of a perfect square trinomial before using factoring. If the equation cannot be written in this form, you may need to use a different method to solve it.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the solution to the equation $3^{2x} - 12 \times 3^x + 27 = 0$ . We have demonstrated how to use factoring, the quadratic formula, and substitution to find the solution for x, and addressed any questions you may have. We hope this article has been helpful in clarifying any confusion you may have had.

Additional Resources

For more information on solving quadratic equations, including those with exponential terms, we recommend the following resources:

We hope this article has provided a helpful guide to solving the equation $3^{2x} - 12 \times 3^x + 27 = 0$ . If you have any questions or need further clarification, please don't hesitate to ask.