Which Is A Solution To $(x-3)(x+9)=-27$?A. X = − 9 X = -9 X = − 9 B. X = − 3 X = -3 X = − 3 C. X = 0 X = 0 X = 0 D. X = 6 X = 6 X = 6

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, $(x-3)(x+9)=-27$, and explore the different methods and techniques used to find the solution.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Given Equation

The given equation is $(x-3)(x+9)=-27$. To solve this equation, we need to isolate the variable $x$. The first step is to expand the left-hand side of the equation using the distributive property.

Expanding the Equation

Using the distributive property, we can expand the left-hand side of the equation as follows:

$$(x-3)(x+9) = x(x+9) - 3(x+9)$$

$$= x^2 + 9x - 3x - 27$$

$$= x^2 + 6x - 27$$

Setting Up the Equation

Now that we have expanded the left-hand side of the equation, we can set it equal to the right-hand side:

$$x^2 + 6x - 27 = -27$$

Simplifying the Equation

To simplify the equation, we can add $27$ to both sides:

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

Factoring the Equation

The next step is to factor the equation. We can factor out the greatest common factor (GCF) of the two terms:

$$x(x + 6) = 0$$

Solving for $x$

Now that we have factored the equation, we can set each factor equal to zero and solve for $x$:

$$x = 0 \quad \text{or} \quad x + 6 = 0$$

$$x = 0 \quad \text{or} \quad x = -6$$

Checking the Solutions

To check the solutions, we can plug each value back into the original equation:

$$(-6-3)(-6+9)=-27$$

$$(-9)(3)=-27$$

$$-27=-27$$

This confirms that $x = -6$ is a solution to the equation.

Conclusion

In this article, we have solved the quadratic equation $(x-3)(x+9)=-27$ using factoring and the distributive property. We have also checked the solutions to confirm that $x = -6$ is a valid solution. This problem illustrates the importance of understanding quadratic equations and the various methods used to solve them.

Comparison of Solutions

Let's compare the solutions to the given equation with the options provided:

A. $x = -9$ B. $x = -3$ C. $x = 0$ D. $x = 6$

The correct solution is $x = -6$, which is not among the options. However, we can see that $x = -9$ is close to the correct solution, but it is not the correct answer.

Final Answer

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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 provide a comprehensive guide to quadratic equations, including a step-by-step solution to the equation $(x-3)(x+9)=-27$. We will also answer some frequently asked questions (FAQs) about quadratic equations.

Q&A: Quadratic Equations

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 (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The method you choose will depend on the specific equation and the level of difficulty.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. The formula is:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the formula and simplify.

Q: What is the difference between factoring and the quadratic formula?

A: Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a mathematical formula to solve the equation.

Q: Can I use both factoring and the quadratic formula to solve a quadratic equation?

A: Yes, you can use both factoring and the quadratic formula to solve a quadratic equation. However, factoring is often a more efficient method, especially for simpler equations.

Q: How do I check my solutions?

A: To check your solutions, plug each value back into the original equation and simplify. If the equation is true, then the value is a valid solution.

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

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

• Not following the order of operations
• Not simplifying the equation correctly
• Not checking the solutions
• Not using the correct method for the specific equation

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

A: Yes, you can use a calculator to solve quadratic equations. However, it's always a good idea to check your solutions by hand to ensure accuracy.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: What are some real-world applications of quadratic equations?

A: 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 economic systems and make predictions about future trends.

Conclusion

In this article, we have provided a comprehensive guide to quadratic equations, including a step-by-step solution to the equation $(x-3)(x+9)=-27$. We have also answered some frequently asked questions (FAQs) about quadratic equations. Whether you're a student or a professional, understanding quadratic equations is essential for success in mathematics and beyond.

Additional Resources

For more information on quadratic equations, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Quadratic Equations

Final Answer

The final answer is $x = -6$.