Solving $2x^2 - 5x + 1 = 3$ A Step-by-Step Guide

Jul 11, 2025 by ADMIN 49 views

Solving Quadratic Equations A Step-by-Step Guide to $2x^2 - 5x + 1 = 3$

Solving quadratic equations is a fundamental skill in algebra, with applications spanning various fields, including physics, engineering, and economics. In this comprehensive guide, we will walk through the process of solving the quadratic equation $2x^2 - 5x + 1 = 3$ , providing a step-by-step explanation and highlighting key concepts along the way. Our goal is to equip you with the knowledge and confidence to tackle similar problems effectively.

Understanding Quadratic Equations

Before diving into the solution, let's first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a ≠ 0. The coefficients a, b, and c determine the shape and position of the parabola represented by the equation. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions correspond to the points where the parabola intersects the x-axis.

Transforming the Equation to Standard Form

To solve the given equation $2x^2 - 5x + 1 = 3$ , our first step is to transform it into the standard form $ax^2 + bx + c = 0$ . This involves moving all terms to one side of the equation, leaving zero on the other side. Subtracting 3 from both sides, we get:

$2x^2 - 5x + 1 - 3 = 0$

Simplifying the equation, we have:

$2x^2 - 5x - 2 = 0$

Now, our equation is in the standard form, with a = 2, b = -5, and c = -2. We are now ready to apply methods for solving quadratic equations.

Methods for Solving Quadratic Equations

There are several methods for solving quadratic equations, each with its advantages and disadvantages. The most common methods include:

Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It is generally the quickest method when applicable, but it is not always straightforward to find the factors.
Completing the Square: This method involves manipulating the equation to form a perfect square trinomial on one side. It is a more general method than factoring and can be used to solve any quadratic equation.
Quadratic Formula: This formula provides a direct solution for x in terms of the coefficients a, b, and c. It is the most versatile method and can be used to solve any quadratic equation, regardless of whether it can be factored.

For the equation $2x^2 - 5x - 2 = 0$ , factoring is not immediately obvious. Therefore, we will focus on the completing the square method and the quadratic formula.

Solving by Completing the Square

Completing the square involves manipulating the quadratic equation to create a perfect square trinomial on one side. This allows us to easily solve for x by taking the square root of both sides. Here are the steps:

Divide by the leading coefficient: If the coefficient of $x^2$ (which is a) is not 1, divide the entire equation by a. In our case, a = 2, so we divide the equation $2x^2 - 5x - 2 = 0$ by 2:

$x^2 - rac{5}{2}x - 1 = 0$
Move the constant term to the right side: Add the constant term to both sides of the equation:

$x^2 - rac{5}{2}x = 1$
Complete the square: To complete the square, we need to add a constant term to both sides of the equation. This constant is equal to the square of half the coefficient of the x term. The coefficient of our x term is -5/2, so half of it is -5/4, and the square of -5/4 is 25/16. Adding 25/16 to both sides:

$x^2 - rac{5}{2}x + rac{25}{16} = 1 + rac{25}{16}$
Factor the left side: The left side is now a perfect square trinomial, which can be factored as:

$(x - rac{5}{4})^2 = 1 + rac{25}{16}$
Simplify the right side: Find a common denominator and add the fractions on the right side:

$(x - rac{5}{4})^2 = rac{16}{16} + rac{25}{16}$

$(x - rac{5}{4})^2 = rac{41}{16}$
Take the square root of both sides:

$x - rac{5}{4} = extstyle rac{\pm \sqrt{41}}{4}$
Solve for x: Add 5/4 to both sides:

$x = rac{5}{4} extstyle \pm \frac{\sqrt{41}}{4}$

Thus, the solutions are $x = rac{5}{4} + rac{\sqrt{41}}{4}$ and $x = rac{5}{4} - rac{\sqrt{41}}{4}$ .

Solving by Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It provides a direct solution for x in terms of the coefficients a, b, and c. The quadratic formula is given by:

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

For our equation $2x^2 - 5x - 2 = 0$ , we have a = 2, b = -5, and c = -2. Plugging these values into the quadratic formula, we get:

$x = rac{-(-5) extstyle \pm \sqrt{(-5)^2 - 4(2)(-2)}}{2(2)}$

Simplifying the expression:

$x = rac{5 extstyle \pm \sqrt{25 + 16}}{4}$

$x = rac{5 extstyle \pm \sqrt{41}}{4}$

So, the solutions are $x = rac{5}{4} + rac{\sqrt{41}}{4}$ and $x = rac{5}{4} - rac{\sqrt{41}}{4}$ , which matches the solutions we found by completing the square.

Conclusion

In this guide, we have demonstrated how to solve the quadratic equation $2x^2 - 5x + 1 = 3$ using two different methods: completing the square and the quadratic formula. Both methods lead to the same solutions: $x = rac{5}{4} extstyle \pm \frac{\sqrt{41}}{4}$ . Understanding these methods is crucial for solving quadratic equations efficiently and accurately. By mastering these techniques, you will be well-equipped to tackle more complex algebraic problems and applications in various fields. Remember to always check your solutions by plugging them back into the original equation to ensure they are correct. Solving quadratic equations is a fundamental skill that opens doors to more advanced mathematical concepts and real-world applications.