Solve For \[$x\$\]:$\[x^2 - 6x - 27 = 0\\]

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $x^2 - 6x - 27 = 0$, and provide a step-by-step guide on how to find the value of $x$. We will also discuss the importance of quadratic equations and their applications in various fields.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving 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. The quadratic formula is a general method for solving quadratic equations, and it can be used to find the solutions of any quadratic equation.

Solving the Given Quadratic Equation

Now, let's apply the quadratic formula to solve the given quadratic equation, $x^2 - 6x - 27 = 0$. We have:

$$a = 1, b = -6, c = -27$$

Substituting these values into the quadratic formula, we get:

$$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-27)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{6 \pm \sqrt{36 + 108}}{2}$$

$$x = \frac{6 \pm \sqrt{144}}{2}$$

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

Therefore, we have two possible solutions:

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

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

Checking the Solutions

To verify that these solutions are correct, we can substitute them back into the original equation. If the equation holds true, then the solution is valid.

For $x = 9$, we have:

$$(9)^2 - 6(9) - 27 = 81 - 54 - 27 = 0$$

This confirms that $x = 9$ is a valid solution.

For $x = -3$, we have:

$$( -3 )^2 - 6( -3 ) - 27 = 9 + 18 - 27 = 0$$

This confirms that $x = -3$ is also a valid solution.

Conclusion

In this article, we have solved a quadratic equation using the quadratic formula. We have also discussed the importance of quadratic equations and their applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of any quadratic equation. By following the steps outlined in this article, you can solve quadratic equations and find the value of $x$.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including:

• Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Computer Science: Quadratic equations are used in algorithms for solving problems, such as finding the shortest path in a graph.
• Economics: Quadratic equations are used to model economic systems, such as supply and demand curves.

Real-World Examples

Quadratic equations have many real-world applications. Here are a few examples:

• Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using quadratic equations.
• Optimization: Quadratic equations can be used to optimize systems, such as finding the minimum or maximum of a function.
• Signal Processing: Quadratic equations are used in signal processing to filter out noise and extract useful information from signals.

Tips and Tricks

Here are a few tips and tricks for solving quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
• Check your solutions: Always check your solutions by substituting them back into the original equation.
• Use algebraic manipulations: Algebraic manipulations, such as factoring and completing the square, can be used to simplify quadratic equations and make them easier to solve.

Conclusion

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will provide answers to frequently asked questions about quadratic equations, including their definition, properties, and methods for solving them.

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 is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are the properties of a quadratic equation?

A: The properties of a quadratic equation include:

• Degree: The degree of a quadratic equation is two.
• Coefficients: The coefficients of a quadratic equation are $a$, $b$, and $c$.
• Variables: The variable of a quadratic equation is $x$.
• Constant term: The constant term of a quadratic equation is $c$.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including:

• Factoring: Factoring involves expressing the quadratic equation as a product of two binomials.
• Completing the square: Completing the square involves rewriting the quadratic equation in the form $(x + p)^2 + q = 0$.
• Quadratic formula: The quadratic formula involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for finding the solutions of a quadratic equation. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to:

• Identify the coefficients: Identify the coefficients $a$, $b$, and $c$ of the quadratic equation.
• Plug in the values: Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
• Simplify the expression: Simplify the expression to find the solutions of the quadratic equation.

Q: What are the solutions of a quadratic equation?

A: The solutions of a quadratic equation are the values of $x$ that satisfy the equation. They can be real or complex numbers.

Q: How do I check my solutions?

A: To check your solutions, you need to:

• Substitute the solutions: Substitute the solutions back into the original equation.
• Check if the equation holds true: Check if the equation holds true for the solutions.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including:

• Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Computer Science: Quadratic equations are used in algorithms for solving problems, such as finding the shortest path in a graph.
• Economics: Quadratic equations are used to model economic systems, such as supply and demand curves.

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

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

• Not checking the solutions: Not checking the solutions to see if they satisfy the equation.
• Not simplifying the expression: Not simplifying the expression to find the solutions of the quadratic equation.
• Not using the correct method: Not using the correct method for solving the quadratic equation.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. By understanding the properties and methods for solving quadratic equations, you can solve them with confidence. Remember to check your solutions and use the correct method for solving the quadratic equation.