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Unraveling the Mystery: Find the Roots of the Following Quadratic Equations If They Exist By The Method Of Completing The Square 2X 2 Plus X Plus 4 0

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Unraveling the Mystery: Find the Roots of the Following Quadratic Equations If They Exist By The Method Of Completing The Square 2X 2 Plus X Plus 4 0

Learn how to find the roots of the quadratic equation 2X^2 + X + 4 = 0 using the completing the square method. Unveil the secrets behind solving this equation effortlessly.

Introduction

In the realm of mathematics, quadratic equations hold a significant place, often encountered in various real-life scenarios. Understanding how to solve them efficiently not only aids in academic endeavors but also nurtures problem-solving skills. Among the diverse methods to solve quadratic equations, completing the square stands out for its versatility and effectiveness. In this comprehensive guide, we delve into the intricacies of finding the roots of the quadratic equation 2X^2 + X + 4 = 0 through the completing the square method.

Exploring Quadratic Equations

Quadratic equations are polynomial equations of the second degree, expressed in the form ax^2 + bx + c = 0, where ‘a’, ‘b’, and ‘c’ are constants and ‘x’ represents the variable. The equation 2X^2 + X + 4 = 0 is a quadratic equation where ‘a’ equals 2, ‘b’ equals 1, and ‘c’ equals 4.

Understanding Completing the Square

Completing the square is a fundamental technique used to solve quadratic equations by transforming them into a perfect square trinomial. This method involves manipulating the equation to express it in the form (x + p)^2 = q, where ‘p’ and ‘q’ are constants.

Deciphering the Equation

To apply the completing the square method to the equation 2X^2 + X + 4 = 0, we follow these steps:

  1. Isolating the Quadratic Term: Begin by isolating the quadratic term, which in this case is 2X^2.

  2. Coefficient Normalization: Normalize the coefficient of the quadratic term to 1 by dividing the entire equation by the coefficient of X^2.

  3. Completing the Square: Complete the square by adding and subtracting the square of half of the coefficient of ‘x’.

  4. Factoring: Factor the perfect square trinomial obtained from completing the square.

  5. Solving for ‘x’: Solve the quadratic equation by taking the square root of both sides and isolating ‘x’.

Implementation of the Method

Let’s illustrate the steps with the equation 2X^2 + X + 4 = 0:

  1. Isolate the Quadratic Term: Subtract 4 from both sides: 2X^2 + X = -4

  2. Coefficient Normalization: Divide the equation by 2: X^2 + 0.5X = -2

  3. Completing the Square: Add and subtract the square of half of the coefficient of ‘x’ (0.25): X^2 + 0.5X + 0.25 = -2 + 0.25 (X + 0.25)^2 = -1.75

  4. Factoring: The perfect square trinomial is (X + 0.25)^2.

  5. Solving for ‘x’: Take the square root of both sides: X + 0.25 = ±√(-1.75)

    Subtract 0.25 from both sides to isolate ‘x’: X = -0.25 ± √(-1.75)

Verification and Validation

To ensure the accuracy of the obtained roots, substitute the values of ‘x’ back into the original equation and verify if they satisfy the equation.

FAQs

  • How do you determine if a quadratic equation has real roots? Quadratic equations have real roots if their discriminant (b^2 – 4ac) is greater than or equal to zero.

  • What is the discriminant of a quadratic equation? The discriminant is the part of the quadratic formula under the square root sign (b^2 – 4ac) and helps determine the nature of roots.

  • Can completing the square method be applied to all quadratic equations? Yes, completing the square method can be applied to all quadratic equations to find their roots.

  • What if the coefficient of the quadratic term is not 1? If the coefficient of the quadratic term is not 1, the equation is first normalized by dividing all terms by the coefficient of the quadratic term.

  • How does completing the square method compare to other methods like factoring and quadratic formula? Completing the square method offers a systematic approach to solving quadratic equations and is particularly useful when factoring or using the quadratic formula is challenging.

  • Are there alternative methods to find the roots of quadratic equations? Yes, besides completing the square, quadratic equations can be solved using methods like factoring, quadratic formula, graphing, and completing the square.

Conclusion

Mastering the art of solving quadratic equations through completing the square method empowers individuals to tackle mathematical challenges with confidence and precision. By unraveling the complexities of equations like 2X^2 + X + 4 = 0, one can enhance their problem-solving skills and foster a deeper understanding of mathematical concepts.

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