Solving The Quadratic Equation X² - 2x + 6 = 0 A Comprehensive Guide

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In the realm of mathematics, quadratic equations hold a significant position. These equations, characterized by their highest power being two, appear in various fields, from physics and engineering to economics and computer science. The equation x2−2x+6=0x^2 - 2x + 6 = 0 presented here is a classic example of a quadratic equation. To effectively tackle such equations, understanding the fundamental concepts and methods is crucial. This article serves as a comprehensive guide to solving quadratic equations, focusing on the given equation as an example. We will delve into the standard form of quadratic equations, explore different solution methods, and provide a step-by-step approach to finding the correct solution.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is expressed as:

ax2+bx+c=0ax^2 + bx + c = 0

where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are also known as roots or zeros, and they represent the values of x that satisfy the equation. These roots can be real or complex numbers.

In the given equation, x2−2x+6=0x^2 - 2x + 6 = 0, we can identify the coefficients as follows:

  • a = 1
  • b = -2
  • c = 6

Understanding these coefficients is essential for applying different methods to solve the equation.

Methods for Solving Quadratic Equations

Several methods can be employed to solve quadratic equations, each with its own advantages and applicability. The most common methods include:

  1. Factoring: This method involves expressing the quadratic equation as a product of two linear factors. It is efficient when the equation can be factored easily.
  2. Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve.
  3. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of its factorability. It is a versatile and widely used method.

For the equation x2−2x+6=0x^2 - 2x + 6 = 0, we will primarily focus on the quadratic formula, as it is the most reliable method for solving any quadratic equation.

1. Factoring

Factoring involves breaking down the quadratic expression into two binomials. For example, the equation x2+5x+6=0x^2 + 5x + 6 = 0 can be factored into (x+2)(x+3)=0(x + 2)(x + 3) = 0. Setting each factor equal to zero gives the solutions x=−2x = -2 and x=−3x = -3. However, not all quadratic equations can be easily factored, especially when the roots are not integers or when the coefficients are complex. In such cases, other methods like completing the square or using the quadratic formula are more appropriate.

The given equation, x2−2x+6=0x^2 - 2x + 6 = 0, does not factor easily using integers. Therefore, we will proceed with other methods to find the solutions.

2. Completing the Square

Completing the square is a technique used to convert a quadratic equation into a perfect square trinomial. This method involves manipulating the equation so that one side becomes a squared binomial. For the general quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the process involves several steps:

  1. Divide the entire equation by a if a is not equal to 1.
  2. Move the constant term (c) to the right side of the equation.
  3. Add the square of half of the coefficient of x (i.e., (b/2)2(b/2)^2) to both sides of the equation.
  4. Rewrite the left side as a squared binomial.
  5. Take the square root of both sides.
  6. Solve for x.

Let's apply this method to the equation x2−2x+6=0x^2 - 2x + 6 = 0:

  1. The coefficient a is already 1, so no division is needed.
  2. Move the constant term to the right side: x2−2x=−6x^2 - 2x = -6.
  3. The coefficient of x is -2, so half of it is -1, and the square of -1 is 1. Add 1 to both sides: x2−2x+1=−6+1x^2 - 2x + 1 = -6 + 1.
  4. Rewrite the left side as a squared binomial: (x−1)2=−5(x - 1)^2 = -5.
  5. Take the square root of both sides: x - 1 = extstyle rac{+}{-} \sqrt{-5}.
  6. Solve for x: x = 1 extstyle rac{+}{-} \sqrt{-5}.

Since we have a negative number under the square root, the solutions are complex numbers. This indicates that the original equation has no real solutions.

3. Quadratic Formula

The quadratic formula is a universally applicable method for solving quadratic equations. It is derived from the process of completing the square and provides a direct way to find the roots of any quadratic equation. The formula is given by:

x = rac{-b extstyle rac{+}{-} extstyle rac{}{} \sqrt{b^2 - 4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.

To solve the equation x2−2x+6=0x^2 - 2x + 6 = 0 using the quadratic formula, we substitute the values of a, b, and c into the formula:

  • a = 1
  • b = -2
  • c = 6

Plugging these values into the quadratic formula, we get:

x = rac{-(-2) extstyle rac{+}{-} extstyle rac{}{} \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}

Simplify the expression step by step:

x = rac{2 extstyle rac{+}{-} extstyle rac{}{} \sqrt{4 - 24}}{2}

x = rac{2 extstyle rac{+}{-} extstyle rac{}{} \sqrt{-20}}{2}

The discriminant, which is the term inside the square root (b2−4acb^2 - 4ac), is -20. Since the discriminant is negative, the solutions will be complex numbers, indicating that there are no real solutions.

Step-by-Step Solution Using the Quadratic Formula

Let's break down the solution process using the quadratic formula for the equation x2−2x+6=0x^2 - 2x + 6 = 0:

  1. Identify the coefficients:
    • a = 1
    • b = -2
    • c = 6
  2. Write down the quadratic formula: x = rac{-b extstyle rac{+}{-} extstyle rac{}{} \sqrt{b^2 - 4ac}}{2a}
  3. Substitute the coefficients into the formula: x = rac{-(-2) extstyle rac{+}{-} extstyle rac{}{} \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}
  4. Simplify the expression: x = rac{2 extstyle rac{+}{-} extstyle rac{}{} \sqrt{4 - 24}}{2} x = rac{2 extstyle rac{+}{-} extstyle rac{}{} \sqrt{-20}}{2}
  5. Analyze the discriminant: The discriminant is -20, which is negative. This indicates that the solutions are complex numbers, and there are no real solutions.
  6. Express the solutions in complex form (if needed): Since −20=20i=25i\sqrt{-20} = \sqrt{20}i = 2\sqrt{5}i, we can write the solutions as: x = rac{2 extstyle rac{+}{-} 2\sqrt{5}i}{2} x = 1 extstyle rac{+}{-} \sqrt{5}i

Thus, the solutions are x=1+5ix = 1 + \sqrt{5}i and x=1−5ix = 1 - \sqrt{5}i.

Analyzing the Discriminant

The discriminant (b2−4acb^2 - 4ac) plays a crucial role in determining the nature of the solutions of a quadratic equation. It provides valuable information about whether the roots are real, complex, distinct, or repeated.

  1. If b2−4ac>0b^2 - 4ac > 0: The equation has two distinct real roots.
  2. If b2−4ac=0b^2 - 4ac = 0: The equation has one real root (a repeated root).
  3. If b2−4ac<0b^2 - 4ac < 0: The equation has two complex roots (no real roots).

For the given equation, x2−2x+6=0x^2 - 2x + 6 = 0, the discriminant is:

b2−4ac=(−2)2−4(1)(6)=4−24=−20b^2 - 4ac = (-2)^2 - 4(1)(6) = 4 - 24 = -20

Since the discriminant is negative (-20), we conclude that the equation has no real solutions, which aligns with our previous calculations.

Choosing the Correct Answer

Based on our analysis, the equation x2−2x+6=0x^2 - 2x + 6 = 0 has no real solutions. Therefore, the correct answer is:

D. no real solution

The other options provide incorrect solutions, either due to errors in applying the quadratic formula or misinterpreting the discriminant.

Common Mistakes and How to Avoid Them

Solving quadratic equations can be challenging, and several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help in avoiding them.

  1. Incorrectly applying the quadratic formula:
    • Ensure that the coefficients a, b, and c are correctly identified.
    • Pay close attention to the signs when substituting values into the formula.
    • Double-check the calculations, especially under the square root.
  2. Making errors in simplification:
    • Simplify the expression step by step to avoid mistakes.
    • Be careful with arithmetic operations, especially when dealing with negative numbers and square roots.
  3. Misinterpreting the discriminant:
    • Understand the relationship between the discriminant and the nature of the roots.
    • If the discriminant is negative, remember that the solutions are complex numbers.
  4. Incorrectly factoring the quadratic equation:
    • If factoring is attempted, ensure that the factors are correctly identified.
    • If the equation does not factor easily, use the quadratic formula or completing the square method.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics. The equation x2−2x+6=0x^2 - 2x + 6 = 0 serves as an excellent example to illustrate the process of using the quadratic formula and interpreting the discriminant. By carefully applying the formula and understanding the nature of the solutions, we can accurately determine that this equation has no real solutions.

In summary, the quadratic formula is a powerful tool for solving any quadratic equation. It is essential to identify the coefficients correctly, substitute them into the formula, and simplify the expression step by step. The discriminant provides valuable insight into the nature of the roots, helping us determine whether they are real or complex. With practice and attention to detail, solving quadratic equations can become a straightforward task.

This guide has provided a comprehensive approach to solving the quadratic equation x2−2x+6=0x^2 - 2x + 6 = 0. By understanding the methods, analyzing the discriminant, and avoiding common mistakes, you can confidently solve similar problems in the future. Whether you are a student, engineer, or anyone dealing with mathematical problems, mastering quadratic equations is a valuable asset.