Solve For { X $}$ In The Equation:${ X^2 - 6x = 0 }$

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Introduction to Solving Quadratic Equations

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: ${ x^2 - 6x = 0 }$. This equation is a quadratic equation in its simplest form, and we will use various methods to solve for the value of $x$.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = -6$, and $c = 0$. To solve for $x$, we need to find the values of $x$ that satisfy the equation.

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 this case, we can factor the equation as follows:

${ x^2 - 6x = x(x - 6) = 0 }$

Using the Zero Product Property

The zero product property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we have:

${ x(x - 6) = 0 }$

This means that either $x = 0$ or $x - 6 = 0$. Solving for $x$ in both cases, we get:

${ x = 0 }$ or ${ x = 6 }$

Checking the Solutions

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

For $x = 0$:

${ (0)^2 - 6(0) = 0 - 0 = 0 }$

This confirms that $x = 0$ is a solution to the equation.

For $x = 6$:

${ (6)^2 - 6(6) = 36 - 36 = 0 }$

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

Conclusion

In this article, we solved the quadratic equation ${ x^2 - 6x = 0 }$ using the factoring method and the zero product property. We found that the solutions to the equation are $x = 0$ and $x = 6$. These solutions can be verified by substituting them back into the original equation.

Additional Tips and Tricks

• When solving quadratic equations, it's essential to check the solutions by substituting them back into the original equation.
• Factoring is a powerful method for solving quadratic equations, but it may not always be possible. In such cases, other methods like the quadratic formula or graphing can be used.
• The zero product property is a fundamental concept in algebra, and it's essential to understand it to solve quadratic equations.

Frequently Asked Questions

• What is a quadratic equation? A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
• How do I solve a quadratic equation? There are several methods for solving quadratic equations, including factoring, the quadratic formula, and graphing.
• What is the zero product property? The zero product property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. In this article, we solved the quadratic equation ${ x^2 - 6x = 0 }$ using the factoring method and the zero product property. We found that the solutions to the equation are $x = 0$ and $x = 6$. These solutions can be verified by substituting them back into the original equation.

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 address some of the most frequently asked questions about quadratic equations and provide detailed answers to help you better understand this topic.

Q1: What is a quadratic equation?

A1: 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 of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q2: How do I solve a quadratic equation?

A2: There are several methods for solving quadratic equations, including:

• Factoring: This involves expressing the quadratic equation as a product of two binomials.
• Quadratic formula: This involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.
• Graphing: This involves graphing the quadratic equation on a coordinate plane and finding the x-intercepts.

Q3: What is the zero product property?

A3: The zero product property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. This property is used to solve quadratic equations by setting each factor equal to zero and solving for the variable.

Q4: How do I factor a quadratic equation?

A4: Factoring a quadratic equation involves expressing it as a product of two binomials. This can be done by:

• Finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
• Writing the quadratic equation as a product of two binomials using these numbers.

Q5: What is the quadratic formula?

A5: The quadratic formula is a formula used to find the solutions of a quadratic equation. It is given by:

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

This formula can be used to find the solutions of a quadratic equation when factoring is not possible.

Q6: How do I use the quadratic formula?

A6: To use the quadratic formula, you need to:

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

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

A7: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a squared variable, while a linear equation does not.

Q8: Can I use the quadratic formula to solve a quadratic equation that cannot be factored?

A8: Yes, you can use the quadratic formula to solve a quadratic equation that cannot be factored. The quadratic formula is a general method for solving quadratic equations, and it can be used in any situation.

Q9: How do I graph a quadratic equation?

A9: To graph a quadratic equation, you need to:

• Identify the values of $a$, $b$, and $c$ in the quadratic equation.
• Use these values to find the x-intercepts and the vertex of the parabola.
• Plot the parabola on a coordinate plane using the x-intercepts and the vertex.

Q10: What is the vertex of a parabola?

A10: The vertex of a parabola is the point on the parabola that is lowest or highest. It is the point where the parabola changes direction.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided detailed answers to help you better understand this topic. Whether you are a student or a professional, we hope that this article has been helpful in your understanding of quadratic equations.