Solving 8x² - 14x + 3 = 0 Quadratic Equation

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In the realm of mathematics, quadratic equations hold a position of prominence, frequently encountered in diverse applications spanning physics, engineering, and computer science. These equations, characterized by the general form ax² + bx + c = 0, where a, b, and c represent constants and a ≠ 0, often present a compelling challenge to solve. This article delves into a detailed exploration of solving the specific quadratic equation 8x² - 14x + 3 = 0, employing various methodologies to unravel its solutions. This comprehensive guide will not only walk you through the step-by-step process but also provide insights into the underlying principles and techniques involved in solving quadratic equations. By the end of this exploration, you will be well-equipped to tackle similar problems with confidence and a deeper understanding of the mathematical concepts involved.

Understanding Quadratic Equations

Before diving into the specifics of solving 8x² - 14x + 3 = 0, it's crucial to establish a solid understanding of quadratic equations in general. A quadratic equation, as mentioned earlier, is defined by the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' cannot be zero. The solutions to a quadratic equation, also known as roots or zeros, represent the values of 'x' that satisfy the equation. These roots correspond to the points where the parabola represented by the equation intersects the x-axis. Quadratic equations can have two distinct real roots, one real root (a repeated root), or two complex roots, depending on the discriminant (b² - 4ac). Understanding these fundamental concepts is essential for navigating the different methods of solving quadratic equations and interpreting the results.

Methods for Solving Quadratic Equations

Several methods are available for solving quadratic equations, each with its own strengths and suitability for different scenarios. The most common methods include:

  1. Factoring: Factoring involves expressing the quadratic expression as a product of two linear factors. This method is efficient when the quadratic expression can be easily factored.
  2. Completing the Square: Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve.
  3. Quadratic Formula: The quadratic formula is a universal method that provides the solutions to any quadratic equation, regardless of its factorability.

Factoring the Quadratic Equation

Let's first attempt to solve 8x² - 14x + 3 = 0 by factoring. The goal is to find two binomials that, when multiplied, yield the original quadratic expression. This involves identifying two numbers that multiply to (8)(3) = 24 and add up to -14. These numbers are -12 and -2. We can then rewrite the middle term (-14x) as -12x - 2x:

8x² - 12x - 2x + 3 = 0

Now, we factor by grouping:

4x(2x - 3) - 1(2x - 3) = 0

(4x - 1)(2x - 3) = 0

Setting each factor equal to zero, we get:

4x - 1 = 0 or 2x - 3 = 0

Solving for x, we find:

x = 1/4 or x = 3/2

Therefore, the solutions to the equation 8x² - 14x + 3 = 0 obtained by factoring are x = 1/4 and x = 3/2.

Completing the Square

Completing the square is another powerful method for solving quadratic equations. This technique involves manipulating the equation to create a perfect square trinomial on one side. Let's apply this method to 8x² - 14x + 3 = 0.

First, divide the entire equation by 8 to make the coefficient of x² equal to 1:

x² - (7/4)x + 3/8 = 0

Next, move the constant term to the right side of the equation:

x² - (7/4)x = -3/8

Now, we need to add a value to both sides of the equation to complete the square. This value is calculated as (b/2a)², where 'b' is the coefficient of the x term and 'a' is the coefficient of the x² term. In this case, b = -7/4 and a = 1, so the value to add is (-7/8)² = 49/64:

x² - (7/4)x + 49/64 = -3/8 + 49/64

The left side of the equation is now a perfect square trinomial:

(x - 7/8)² = -24/64 + 49/64

(x - 7/8)² = 25/64

Take the square root of both sides:

x - 7/8 = ±√(25/64)

x - 7/8 = ±5/8

Solve for x:

x = 7/8 ± 5/8

x = (7 + 5)/8 = 12/8 = 3/2 or x = (7 - 5)/8 = 2/8 = 1/4

As we can see, completing the square yields the same solutions as factoring: x = 1/4 and x = 3/2.

Using the Quadratic Formula

The quadratic formula is a universally applicable method for solving any quadratic equation of the form ax² + bx + c = 0. The formula is given by:

x = [-b ± √(b² - 4ac)] / 2a

For our equation 8x² - 14x + 3 = 0, we have a = 8, b = -14, and c = 3. Plugging these values into the quadratic formula, we get:

x = [14 ± √((-14)² - 4 * 8 * 3)] / (2 * 8)

x = [14 ± √(196 - 96)] / 16

x = [14 ± √100] / 16

x = [14 ± 10] / 16

Solving for the two possible values of x:

x = (14 + 10) / 16 = 24 / 16 = 3/2

x = (14 - 10) / 16 = 4 / 16 = 1/4

Once again, the quadratic formula confirms the solutions as x = 1/4 and x = 3/2. This demonstrates the power and reliability of the quadratic formula in solving quadratic equations.

Conclusion

In conclusion, we have successfully solved the quadratic equation 8x² - 14x + 3 = 0 using three distinct methods: factoring, completing the square, and the quadratic formula. All three methods consistently yielded the same solutions, x = 1/4 and x = 3/2. This exercise highlights the versatility of mathematical techniques and the importance of understanding different approaches to problem-solving. Whether you prefer the elegance of factoring, the systematic approach of completing the square, or the universal applicability of the quadratic formula, having a repertoire of methods at your disposal allows you to tackle a wide range of quadratic equations with confidence. Furthermore, this exploration reinforces the fundamental principles of quadratic equations and their solutions, providing a solid foundation for further mathematical endeavors. Understanding how to solve quadratic equations is a crucial skill in various fields, from mathematics and physics to engineering and computer science, making it an invaluable asset in any problem-solving toolkit. Remember to practice these techniques with various examples to solidify your understanding and enhance your proficiency in solving quadratic equations. By mastering these methods, you will be well-equipped to tackle more complex mathematical challenges and apply these principles to real-world problems.