Solve $3^2 X - 6(3 X) + 5 = 0$.

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Introduction

In this article, we will be solving a quadratic equation of the form $3^2 x - 6(3 x) + 5 = 0$. This equation is a quadratic equation because it can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The equation we are given is a quadratic equation in one variable, and we will be using algebraic methods to solve it.

Understanding the Equation

The given equation is $3^2 x - 6(3 x) + 5 = 0$. To simplify this equation, we can start by evaluating the exponent $3^2$, which is equal to $9$. We can then rewrite the equation as $9x - 6(3x) + 5 = 0$.

Distributing the Negative 6

To simplify the equation further, we can distribute the negative 6 to the term $3x$. This gives us $9x - 18x + 5 = 0$.

Combining Like Terms

We can now combine the like terms $9x$ and $-18x$, which are both multiples of $x$. This gives us $-9x + 5 = 0$.

Isolating the Variable

To isolate the variable $x$, we can add $9x$ to both sides of the equation. This gives us $5 = 9x$.

Solving for x

To solve for $x$, we can divide both sides of the equation by $9$. This gives us $x = \frac{5}{9}$.

Conclusion

In this article, we have solved the quadratic equation $3^2 x - 6(3 x) + 5 = 0$. We started by simplifying the equation, distributing the negative 6, combining like terms, isolating the variable, and finally solving for $x$. The solution to the equation is $x = \frac{5}{9}$.

Final Answer

The final answer to the equation $3^2 x - 6(3 x) + 5 = 0$ is $x = \frac{5}{9}$.

Step-by-Step Solution

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

  1. Simplify the equation: $3^2 x - 6(3 x) + 5 = 0$
  2. Distribute the negative 6: $9x - 18x + 5 = 0$
  3. Combine like terms: $-9x + 5 = 0$
  4. Add $9x$ to both sides: $5 = 9x$
  5. Divide both sides by $9$: $x = \frac{5}{9}$

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

  • Simplify the equation as much as possible before trying to solve it.
  • Use the distributive property to distribute negative numbers.
  • Combine like terms to simplify the equation.
  • Isolate the variable by adding or subtracting the same value from both sides.
  • Divide both sides of the equation by a non-zero value to solve for the variable.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

  • Not simplifying the equation enough before trying to solve it.
  • Not using the distributive property to distribute negative numbers.
  • Not combining like terms to simplify the equation.
  • Not isolating the variable by adding or subtracting the same value from both sides.
  • Not dividing both sides of the equation by a non-zero value to solve for the variable.

Real-World Applications

Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.
  • Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Conclusion

In conclusion, solving quadratic equations is an important skill that has many real-world applications. By following the steps outlined in this article, you can solve quadratic equations and apply them to a variety of fields. Remember to simplify the equation, distribute negative numbers, combine like terms, isolate the variable, and divide both sides of the equation by a non-zero value to solve for the variable.

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Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

  1. Simplify the equation as much as possible.
  2. Use the distributive property to distribute negative numbers.
  3. Combine like terms to simplify the equation.
  4. Isolate the variable by adding or subtracting the same value from both sides.
  5. Divide both sides of the equation by a non-zero value to solve for the variable.

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. It can be written in the form ax + b = 0, where a and b are constants. A quadratic equation, on the other hand, is a polynomial equation of degree two.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula that you can use to solve the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. You can then simplify the expression and solve for x.

Q: What are the different types of solutions to a quadratic equation?

A: There are three different types of solutions to a quadratic equation:

  1. Real and distinct solutions: These are solutions that are real numbers and are distinct from each other.
  2. Real and repeated solutions: These are solutions that are real numbers and are repeated.
  3. Complex solutions: These are solutions that are complex numbers.

Q: How do I determine the type of solution to a quadratic equation?

A: To determine the type of solution to a quadratic equation, you can use the discriminant, which is given by b^2 - 4ac. If the discriminant is positive, then the solutions are real and distinct. If the discriminant is zero, then the solutions are real and repeated. If the discriminant is negative, then the solutions are complex.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

  • Not simplifying the equation enough before trying to solve it.
  • Not using the distributive property to distribute negative numbers.
  • Not combining like terms to simplify the equation.
  • Not isolating the variable by adding or subtracting the same value from both sides.
  • Not dividing both sides of the equation by a non-zero value to solve for the variable.

Q: How do I apply quadratic equations to real-world problems?

A: Quadratic equations can be applied to a variety of real-world problems, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.
  • Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Q: What are some advanced topics in quadratic equations?

A: Some advanced topics in quadratic equations include:

  • Quadratic equations with complex coefficients.
  • Quadratic equations with irrational coefficients.
  • Quadratic equations with multiple variables.
  • Quadratic equations with non-linear terms.

Q: How do I learn more about quadratic equations?

A: There are many resources available to learn more about quadratic equations, including:

  • Textbooks and online resources.
  • Video tutorials and online courses.
  • Practice problems and quizzes.
  • Real-world applications and examples.