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 a specific quadratic equation, $x^2 - 4x = 2$, and provide a step-by-step guide on how to find the value of x.

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, 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:

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

This formula can be used to find the solutions to any quadratic equation in the form $ax^2 + bx + c = 0$.

Solving the Given Quadratic Equation

Now, let's apply the quadratic formula to solve the given quadratic equation, $x^2 - 4x = 2$. To do this, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting 2 from both sides of the equation:

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

Now, we can identify the values of a, b, and c:

$$a = 1, b = -4, c = -2$$

Applying the Quadratic Formula

Substituting the values of a, b, and c 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 solved the quadratic equation $x^2 - 4x = 2$ using the quadratic formula. We have shown that the solutions to this equation are $x = 2 \pm \sqrt{6}$. This demonstrates the power of the quadratic formula in solving quadratic equations.

Tips and Tricks

• When solving quadratic equations, make sure to rewrite the equation in the standard form $ax^2 + bx + c = 0$.
• Use the quadratic formula to find the solutions to the equation.
• Simplify the expression to get the final solutions.

Real-World Applications

Quadratic equations have numerous 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.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to solve quadratic equations, we can unlock the secrets of mathematics and make predictions about the world around us. Whether you are a student or a professional, mastering the art of solving quadratic equations will serve you well in your future endeavors.

Additional Resources

For further learning, we recommend the following resources:

• Khan Academy: Quadratic Equations
• MIT OpenCourseWare: Quadratic Equations
• Wolfram Alpha: Quadratic Equations

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 (in this case, x) is two.

Q: How do I solve a quadratic equation? A: You can solve a quadratic equation using various methods, including factoring, completing the square, and the quadratic formula.

Q: What is the quadratic formula? A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

Q: How do I apply the quadratic formula? A: To apply the quadratic formula, you need to identify the values of a, b, and c in the equation, and then substitute them into the formula.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to solve quadratic equations, we can unlock the secrets of mathematics and make predictions about the world around us. Whether you are a student or a professional, mastering the art of solving quadratic equations will serve you well in your future endeavors.

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 Q&A guide on quadratic equations, covering frequently asked questions and answers.

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: You can solve a quadratic equation using various methods, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of a, b, and c in the equation, and then substitute them into the formula. For example, if you have the equation $x^2 + 4x + 4 = 0$, you can identify the values of a, b, and c as follows:

$$a = 1, b = 4, c = 4$$

Substituting these values into the quadratic formula, you get:

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

Simplifying the expression, you get:

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

$$x = \frac{-4 \pm \sqrt{0}}{2}$$

$$x = \frac{-4}{2}$$

$$x = -2$$

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. The general form of a linear equation is $ax + b = 0$, where a and b are constants.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring. Factoring involves expressing the quadratic equation as a product of two binomials. For example, if you have the equation $x^2 + 5x + 6 = 0$, you can factor it as follows:

$$(x + 3)(x + 2) = 0$$

This tells you that either $x + 3 = 0$ or $x + 2 = 0$. Solving for x, you get:

$$x = -3$$

$$x = -2$$

Q: Can I solve a quadratic equation by completing the square?

A: Yes, you can solve a quadratic equation by completing the square. Completing the square involves rewriting the quadratic equation in the form $(x + p)^2 = q$, where p and q are constants.

Q: What is the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$. The discriminant determines the nature of the solutions to the quadratic equation.

Q: What do the values of the discriminant tell me?

A: The values of the discriminant tell you the nature of the solutions to the quadratic 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 a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you the complex solutions to the equation.

Q: How do I simplify the expression under the square root in the quadratic formula?

A: To simplify the expression under the square root in the quadratic formula, you need to factor the expression and simplify it. For example, if you have the expression $b^2 - 4ac$, you can factor it as follows:

$$(b - 2a)(b + 2a)$$

Simplifying the expression, you get:

$$b^2 - 4ac = (b - 2a)(b + 2a)$$

Q: Can I use the quadratic formula to solve a quadratic equation with rational solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with rational solutions. The quadratic formula will give you the rational solutions to the equation.

Q: How do I apply the quadratic formula to solve a quadratic equation with rational solutions?

A: To apply the quadratic formula to solve a quadratic equation with rational solutions, you need to identify the values of a, b, and c in the equation, and then substitute them into the formula. For example, if you have the equation $x^2 + 4x + 4 = 0$, you can identify the values of a, b, and c as follows:

$$a = 1, b = 4, c = 4$$

Substituting these values into the quadratic formula, you get:

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

Simplifying the expression, you get:

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

$$x = \frac{-4 \pm \sqrt{0}}{2}$$

$$x = \frac{-4}{2}$$

$$x = -2$$

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to solve quadratic equations, we can unlock the secrets of mathematics and make predictions about the world around us. Whether you are a student or a professional, mastering the art of solving quadratic equations will serve you well in your future endeavors.