Select The Correct Answer.Solve The Following Equation For X X X . 12 X 2 − 36 X = 0 12x^2 - 36x = 0 12 X 2 − 36 X = 0 A. X = 0 , 3 X = 0, 3 X = 0 , 3 B. X = 0 , 1 3 X = 0, \frac{1}{3} X = 0 , 3 1 C. X = 0 , − 3 X = 0, -3 X = 0 , − 3 D. X = 1 4 , 3 X = \frac{1}{4}, 3 X = 4 1 , 3

Mar 12, 2025 by ADMIN 284 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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, $12x^2 - 36x = 0$ , and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. In our given equation, $12x^2 - 36x = 0$ , we can identify the coefficients as $a = 12$ , $b = -36$ , and $c = 0$ .

Factoring the Equation

One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In our equation, we can factor out the greatest common factor (GCF) of the two terms, which is $12x$ . Factoring out $12x$ gives us:

12x^2 - 36x = 12x(x - 3)

Now, we can see that the equation can be factored as:

12x(x - 3) = 0

Solving for $x$

To solve for $x$ , we need to find the values of $x$ that make the equation true. In this case, we have two factors: $12x$ and $(x - 3)$ . We can set each factor equal to zero and solve for $x$ :

12x = 0 \Rightarrow x = 0

(x - 3) = 0 \Rightarrow x = 3

Therefore, the solutions to the equation are $x = 0$ and $x = 3$ .

Checking the Solutions

To verify that our solutions are correct, we can plug them back into the original equation:

12(0)^2 - 36(0) = 0

12(3)^2 - 36(3) = 0

Both equations are true, which confirms that our solutions are correct.

Conclusion

In this article, we solved the quadratic equation $12x^2 - 36x = 0$ using the factoring method. We factored out the GCF of the two terms, set each factor equal to zero, and solved for $x$ . The solutions to the equation are $x = 0$ and $x = 3$ . We also verified our solutions by plugging them back into the original equation.

Answer

The correct answer is:

A. $x = 0, 3$

This is the only option that matches our solutions.

Discussion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we focused on solving a specific quadratic equation, $12x^2 - 36x = 0$ , and explored the different methods and techniques used to find the solutions. We also discussed the importance of verifying our solutions by plugging them back into the original equation.

References

"Algebra and Trigonometry" by Michael Sullivan
"College Algebra" by James Stewart
"Quadratic Equations" by Math Open Reference
Quadratic Equations Q&A ==========================

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 is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, using the quadratic formula, and graphing. The method you choose will depend on the specific equation and the type of solution you are looking for.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations that involves using the coefficients of the equation to find the solutions. The quadratic 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 coefficients of the equation, which are $a$ , $b$ , and $c$ . You then plug these values into the quadratic formula and simplify to find the solutions.

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

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using the coefficients of the equation to find the solutions.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation. However, it may not always be the most efficient method, especially for equations that can be easily factored.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to identify the solutions to the equation and plot them on a coordinate plane. You can also use a graphing calculator or software to graph the equation.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$ . The discriminant determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can I use the quadratic formula to solve systems of quadratic equations?

A: Yes, the quadratic formula can be used to solve systems of quadratic equations. However, it may not always be the most efficient method, especially for systems with multiple equations.

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

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

Not identifying the coefficients of the equation correctly
Not simplifying the quadratic formula correctly
Not checking the solutions to the equation
Not using the correct method for solving the equation

Q: How do I choose the correct method for solving a quadratic equation?

A: To choose the correct method for solving a quadratic equation, you need to consider the type of equation and the type of solution you are looking for. If the equation can be easily factored, factoring may be the best method. If the equation cannot be easily factored, the quadratic formula may be a better option.

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 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: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including:

Algebra: Quadratic equations can be used to solve systems of linear equations.
Geometry: Quadratic equations can be used to find the area and perimeter of shapes.
Trigonometry: Quadratic equations can be used to solve trigonometric equations.

Q: How do I practice solving quadratic equations?

A: To practice solving quadratic equations, you can:

Work through examples and exercises in a textbook or online resource.
Use a graphing calculator or software to graph quadratic equations.
Solve quadratic equations in real-world applications, such as physics or engineering.
Join a study group or online community to practice solving quadratic equations with others.