Solve For \[$ X \$\]:$\[ X^2 - 4x = 2 \\]

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 quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the given equation $x^2 - 4x = 2$ as an example to demonstrate the steps involved in solving quadratic equations.

Understanding the Basics of Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable $x$ is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The value of $a$ cannot be zero, as this would result in a linear equation rather than a quadratic one.

Key Components of Quadratic Equations

• Coefficient of $x^2$: The coefficient of $x^2$ is denoted by $a$. In the given equation, $a = 1$.
• Coefficient of $x$: The coefficient of $x$ is denoted by $b$. In the given equation, $b = -4$.
• Constant Term: The constant term is denoted by $c$. In the given equation, $c = -2$.

Rearranging the Given Equation

To solve the given equation $x^2 - 4x = 2$, we need to rearrange it to the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. We can do this by subtracting $2$ from both sides of the equation:

$$x^2 - 4x - 2 = 0$$

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

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

In our case, $a = 1$, $b = -4$, and $c = -2$. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

$$x = \frac{4 \pm \sqrt{16 + 8}}{2}$$

$$x = \frac{4 \pm \sqrt{24}}{2}$$

$$x = \frac{4 \pm 2\sqrt{6}}{2}$$

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

Conclusion

In this article, we have demonstrated the steps involved in solving quadratic equations using the quadratic formula. We have used the given equation $x^2 - 4x = 2$ as an example to illustrate the process. By following these steps, you can solve quadratic equations of the form $ax^2 + bx + c = 0$ and find the value of $x$.

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 $x$ is two.
• How do I solve a quadratic equation? You can solve a quadratic equation using the quadratic formula, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Further Reading

• Quadratic Equations: A Comprehensive Guide This article provides a comprehensive guide to quadratic equations, including the quadratic formula, factoring, and graphing.
• Solving Quadratic Equations: A Step-by-Step Guide This article provides a step-by-step guide to solving quadratic equations using the quadratic formula.
• Quadratic Equations: Applications and Examples This article provides examples and applications of quadratic equations in real-world scenarios.

References

• "Quadratic Equations" by Math Open Reference
• "Quadratic Formula" by Khan Academy
• "Solving Quadratic Equations" by Purplemath
  

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, solving methods, and applications.

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 $x$ 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: You can solve a quadratic equation using the quadratic formula, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Alternatively, you can factor the quadratic expression or use the method of completing the square.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a highest power of $x$ equal to two, while a linear equation has a highest power of $x$ equal to one.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if the quadratic expression can be expressed as a product of two binomials. For example, the quadratic expression $x^2 + 5x + 6$ can be factored as $(x + 3)(x + 2)$.

Q: What is the method of completing the square?

A: The method of completing the square is a technique for solving quadratic equations by rewriting the quadratic expression in the form $(x + p)^2 = q$. This method is useful for solving quadratic equations that cannot be factored easily.

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 function that can be used to solve quadratic equations.

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

A: Quadratic equations have many real-world applications, including physics, engineering, economics, and computer science. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation.

Conclusion

In this article, we have provided answers to frequently asked questions about quadratic equations, including their definition, solving methods, and applications. We hope that this article has been helpful in clarifying any doubts you may have had about 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 $x$ is two.
• How do I solve a quadratic equation? You can solve a quadratic equation using the quadratic formula, factoring, or the method of completing the square.
• What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Further Reading

• Quadratic Equations: A Comprehensive Guide This article provides a comprehensive guide to quadratic equations, including the quadratic formula, factoring, and graphing.
• Solving Quadratic Equations: A Step-by-Step Guide This article provides a step-by-step guide to solving quadratic equations using the quadratic formula.
• Quadratic Equations: Applications and Examples This article provides examples and applications of quadratic equations in real-world scenarios.

References

• "Quadratic Equations" by Math Open Reference
• "Quadratic Formula" by Khan Academy
• "Solving Quadratic Equations" by Purplemath