Solve The Equation:${ 3^{2x+1} - 26 \left(3^x\right) - 9 = 0 }$

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Introduction

In this article, we will delve into the world of mathematics and explore a method to solve a quadratic equation in the form of $3^{2x+1} - 26 \left(3^x\right) - 9 = 0$. This equation may seem daunting at first, but with the right approach, we can break it down and find the solution. We will use algebraic manipulation and substitution to simplify the equation and ultimately find the value of $x$.

Understanding the Equation

The given equation is a quadratic equation in the form of $a^2x + a + b = 0$, where $a = 3^x$ and $b = -9$. However, this equation is not in the standard quadratic form, as it involves exponentiation. To make it more manageable, we can use a substitution to simplify the equation.

Substitution Method

Let's substitute $y = 3^x$ into the equation. This will allow us to rewrite the equation in terms of $y$ and simplify it. The equation becomes:

$$y^2 - 26y - 9 = 0$$

This is a standard quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = -26$, and $c = -9$. We can use the quadratic formula to solve for $y$:

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

Quadratic Formula

Substituting the values of $a$, $b$, and $c$ into the quadratic formula, we get:

$$y = \frac{-(-26) \pm \sqrt{(-26)^2 - 4(1)(-9)}}{2(1)}$$

Simplifying the expression, we get:

$$y = \frac{26 \pm \sqrt{676 + 36}}{2}$$

$$y = \frac{26 \pm \sqrt{712}}{2}$$

$$y = \frac{26 \pm 26.6}{2}$$

Solving for $y$

We have two possible values for $y$:

$$y = \frac{26 + 26.6}{2} = 26.3$$

$$y = \frac{26 - 26.6}{2} = -0.3$$

Substituting Back

Now that we have the values of $y$, we can substitute back to find the values of $x$. Recall that $y = 3^x$, so we can rewrite the equation as:

$$3^x = 26.3$$

$$3^x = -0.3$$

Solving for $x$

To solve for $x$, we can take the logarithm of both sides of the equation. We will use the natural logarithm (ln) for this purpose:

$$\ln(3^x) = \ln(26.3)$$

$$\ln(3^x) = \ln(-0.3)$$

Using the property of logarithms that $\ln(a^b) = b\ln(a)$, we can simplify the equation:

$$x\ln(3) = \ln(26.3)$$

$$x\ln(3) = \ln(-0.3)$$

Solving for $x$

Now that we have the equation in the form of $x\ln(3) = \ln(26.3)$, we can solve for $x$:

$$x = \frac{\ln(26.3)}{\ln(3)}$$

$$x = \frac{\ln(-0.3)}{\ln(3)}$$

Calculating the Values

Using a calculator to evaluate the expressions, we get:

$$x = \frac{\ln(26.3)}{\ln(3)} = 4.4$$

$$x = \frac{\ln(-0.3)}{\ln(3)} = -1.1$$

Conclusion

In this article, we used the substitution method to simplify the equation $3^{2x+1} - 26 \left(3^x\right) - 9 = 0$. We then used the quadratic formula to solve for $y$ and substituted back to find the values of $x$. The final values of $x$ are $4.4$ and $-1.1$. This method demonstrates the power of algebraic manipulation and substitution in solving complex equations.

Final Thoughts

Solving equations like $3^{2x+1} - 26 \left(3^x\right) - 9 = 0$ requires patience, persistence, and a deep understanding of algebraic manipulation and substitution. By breaking down the equation into manageable parts and using the quadratic formula, we can find the solution to the equation. This method can be applied to a wide range of mathematical problems, from simple algebra to complex calculus.

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Q: What is the main concept behind solving the equation $3^{2x+1} - 26 \left(3^x\right) - 9 = 0$?

A: The main concept behind solving this equation is to use algebraic manipulation and substitution to simplify the equation and ultimately find the value of $x$. We use the substitution method to rewrite the equation in terms of a new variable, $y$, and then use the quadratic formula to solve for $y$. Finally, we substitute back to find the values of $x$.

Q: What is the quadratic formula, and how is it used to solve the equation?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

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

We use this formula to solve for $y$ in the equation $y^2 - 26y - 9 = 0$.

Q: What is the significance of the natural logarithm (ln) in solving the equation?

A: The natural logarithm (ln) is used to solve for $x$ in the equation $x\ln(3) = \ln(26.3)$. The property of logarithms that $\ln(a^b) = b\ln(a)$ allows us to simplify the equation and solve for $x$.

Q: Can you explain the concept of substitution in solving the equation?

A: Substitution is a mathematical technique used to simplify an equation by replacing a variable with a new variable. In this case, we substitute $y = 3^x$ into the equation to rewrite it in terms of $y$. This allows us to use the quadratic formula to solve for $y$ and ultimately find the values of $x$.

Q: What are the final values of $x$ that solve the equation?

A: The final values of $x$ that solve the equation are $4.4$ and $-1.1$.

Q: Can you provide a step-by-step guide to solving the equation?

A: Here is a step-by-step guide to solving the equation:

1. Substitute $y = 3^x$ into the equation to rewrite it in terms of $y$.
2. Use the quadratic formula to solve for $y$ in the equation $y^2 - 26y - 9 = 0$.
3. Substitute back to find the values of $x$.
4. Use the natural logarithm (ln) to solve for $x$ in the equation $x\ln(3) = \ln(26.3)$.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

• Not using the correct substitution method
• Not using the quadratic formula correctly
• Not simplifying the equation correctly
• Not using the natural logarithm (ln) correctly

Q: Can you provide additional resources for learning more about solving equations like this?

A: Yes, here are some additional resources for learning more about solving equations like this:

• Algebra textbooks and online resources
• Math websites and online communities
• Calculators and software for solving equations
• Online courses and tutorials for learning more about algebra and mathematics.

Q: Is there a way to check the solutions to the equation?

A: Yes, there are several ways to check the solutions to the equation, including:

• Plugging the solutions back into the original equation to see if they are true
• Using a calculator or software to verify the solutions
• Checking the solutions against the original equation to see if they are consistent.

Q: Can you provide a summary of the key concepts and steps involved in solving the equation?

A: Here is a summary of the key concepts and steps involved in solving the equation:

• Substitution method: substitute $y = 3^x$ into the equation to rewrite it in terms of $y$
• Quadratic formula: use the quadratic formula to solve for $y$ in the equation $y^2 - 26y - 9 = 0$
• Natural logarithm (ln): use the natural logarithm (ln) to solve for $x$ in the equation $x\ln(3) = \ln(26.3)$
• Simplification: simplify the equation correctly to find the values of $x$.

I hope this Q&A article has been helpful in providing additional information and resources for solving the equation $3^{2x+1} - 26 \left(3^x\right) - 9 = 0$.