Solve For { X $}$ In The Equation:${ 3 {x 2+6}=9^{2 \cdot 5 X} }$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving the equation $3^{x^2+6}=9^{2 \cdot 5 x}$, which involves manipulating exponents and using logarithmic properties to isolate the variable x.

Understanding Exponential Equations

Exponential equations are equations that involve exponents, which are numbers that represent the power to which a base number is raised. In the equation $3^{x^2+6}=9^{2 \cdot 5 x}$, the base numbers are 3 and 9, and the exponents are $x^2+6$ and $2 \cdot 5 x$, respectively.

Manipulating Exponents

To solve the equation, we need to manipulate the exponents to isolate the variable x. We can start by rewriting the equation using the fact that $9 = 3^2$. This gives us:

$$3^{x^2+6} = (3^2)^{2 \cdot 5 x}$$

Using the property of exponents that $(a^b)^c = a^{bc}$, we can rewrite the equation as:

$$3^{x^2+6} = 3^{2 \cdot 5 x \cdot 2}$$

Equating Exponents

Since the bases are the same, we can equate the exponents:

$$x^2+6 = 2 \cdot 5 x \cdot 2$$

Simplifying the equation, we get:

$$x^2+6 = 20 x$$

Rearranging the Equation

To isolate the variable x, we need to rearrange the equation. We can start by subtracting 20x from both sides:

$$x^2 - 20 x + 6 = 0$$

Using the Quadratic Formula

The equation is a quadratic equation, and we can use the quadratic formula to solve for x:

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

In this case, a = 1, b = -20, and c = 6. Plugging these values into the formula, we get:

$$x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(1)(6)}}{2(1)}$$

Simplifying the equation, we get:

$$x = \frac{20 \pm \sqrt{400 - 24}}{2}$$

$$x = \frac{20 \pm \sqrt{376}}{2}$$

Simplifying the Square Root

The square root of 376 can be simplified as:

$$\sqrt{376} = \sqrt{4 \cdot 94} = 2 \sqrt{94}$$

Substituting this value back into the equation, we get:

$$x = \frac{20 \pm 2 \sqrt{94}}{2}$$

Simplifying the Equation

Simplifying the equation further, we get:

$$x = 10 \pm \sqrt{94}$$

Conclusion

In this article, we have solved the exponential equation $3^{x^2+6}=9^{2 \cdot 5 x}$ using algebraic manipulations and properties of exponents. We have used the quadratic formula to solve for x and simplified the square root to obtain the final solution. The solution to the equation is $x = 10 \pm \sqrt{94}$.

Final Answer

The final answer is $\boxed{10 \pm \sqrt{94}}$.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. Rewrite the equation using the fact that $9 = 3^2$.
2. Use the property of exponents that $(a^b)^c = a^{bc}$ to rewrite the equation.
3. Equate the exponents since the bases are the same.
4. Simplify the equation and rearrange it to isolate the variable x.
5. Use the quadratic formula to solve for x.
6. Simplify the square root and substitute the value back into the equation.
7. Simplify the equation further to obtain the final solution.

Frequently Asked Questions

• What is an exponential equation? An exponential equation is an equation that involves exponents, which are numbers that represent the power to which a base number is raised.
• How do I solve an exponential equation? To solve an exponential equation, you need to manipulate the exponents to isolate the variable x. You can use algebraic manipulations and properties of exponents to simplify the equation and solve for x.
• What is the quadratic formula? The quadratic formula is a formula that is used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• How do I simplify a square root? To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. You can then rewrite the square root as the product of the perfect square and the remaining number.

Related Topics

• Exponential functions
• Logarithmic functions
• Quadratic equations
• Algebraic manipulations
• Properties of exponents

References

• [1] "Exponential Equations" by Math Open Reference
• [2] "Quadratic Formula" by Math Is Fun
• [3] "Simplifying Square Roots" by Khan Academy
  

Introduction

Exponential equations can be challenging to solve, but with the right approach and techniques, you can master them. In this article, we will answer some of the most frequently asked questions about exponential equations, providing you with a deeper understanding of these equations and how to solve them.

Q1: What is an exponential equation?

A1: An exponential equation is an equation that involves exponents, which are numbers that represent the power to which a base number is raised. Exponential equations can be written in the form $a^x = b$, where a is the base, x is the exponent, and b is the result.

Q2: How do I solve an exponential equation?

A2: To solve an exponential equation, you need to manipulate the exponents to isolate the variable x. You can use algebraic manipulations and properties of exponents to simplify the equation and solve for x. One common technique is to use the fact that $a^x = b$ can be rewritten as $x = \log_a b$, where $\log_a b$ is the logarithm of b to the base a.

Q3: What is the difference between an exponential equation and a logarithmic equation?

A3: An exponential equation is an equation that involves exponents, while a logarithmic equation is an equation that involves logarithms. Exponential equations can be written in the form $a^x = b$, while logarithmic equations can be written in the form $x = \log_a b$.

Q4: How do I simplify an exponential equation?

A4: To simplify an exponential equation, you can use algebraic manipulations and properties of exponents. For example, you can use the fact that $a^x = a^{x+y}$ to combine exponents, or use the fact that $a^x = (a^y)^{x/y}$ to simplify fractions.

Q5: What is the quadratic formula?

A5: The quadratic formula is a formula that is used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic equation.

Q6: How do I use the quadratic formula to solve an exponential equation?

A6: To use the quadratic formula to solve an exponential equation, you need to rewrite the equation in the form $ax^2 + bx + c = 0$, where a, b, and c are the coefficients of the quadratic equation. You can then plug these values into the quadratic formula to solve for x.

Q7: What is the difference between a linear equation and an exponential equation?

A7: A linear equation is an equation that involves a linear function, while an exponential equation is an equation that involves an exponential function. Linear equations can be written in the form $y = mx + b$, while exponential equations can be written in the form $y = a^x$.

Q8: How do I graph an exponential equation?

A8: To graph an exponential equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot the graph.

Q9: What is the domain of an exponential equation?

A9: The domain of an exponential equation is the set of all possible values of x for which the equation is defined. For example, if the equation is $a^x = b$, the domain is all real numbers x.

Q10: What is the range of an exponential equation?

A10: The range of an exponential equation is the set of all possible values of y for which the equation is defined. For example, if the equation is $a^x = b$, the range is all positive real numbers y.

Conclusion

Exponential equations can be challenging to solve, but with the right approach and techniques, you can master them. In this article, we have answered some of the most frequently asked questions about exponential equations, providing you with a deeper understanding of these equations and how to solve them.

Final Answer

The final answer is that exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. Rewrite the equation using the fact that $a^x = b$ can be rewritten as $x = \log_a b$.
2. Use the quadratic formula to solve for x.
3. Simplify the equation and solve for x.
4. Use algebraic manipulations and properties of exponents to simplify the equation.
5. Graph the equation using a graphing calculator or a computer program.
6. Find the domain and range of the equation.

Frequently Asked Questions

• What is an exponential equation?
• How do I solve an exponential equation?
• What is the difference between an exponential equation and a logarithmic equation?
• How do I simplify an exponential equation?
• What is the quadratic formula?
• How do I use the quadratic formula to solve an exponential equation?
• What is the difference between a linear equation and an exponential equation?
• How do I graph an exponential equation?
• What is the domain of an exponential equation?
• What is the range of an exponential equation?

Related Topics

• Exponential functions
• Logarithmic functions
• Quadratic equations
• Algebraic manipulations
• Properties of exponents

References

• [1] "Exponential Equations" by Math Open Reference
• [2] "Quadratic Formula" by Math Is Fun
• [3] "Simplifying Exponential Equations" by Khan Academy