Solving 2x^2 - 9x + 7 = 0: A Step-by-Step Guide

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Hey guys! Let's dive into solving this quadratic equation: 2x^2 - 9x + 7 = 0. Quadratic equations might seem intimidating at first, but don't worry! We'll break it down step by step so you can conquer any quadratic equation that comes your way. In this comprehensive guide, we'll explore the various methods to find the values of 'x' that satisfy this equation. So, buckle up, and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is: ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we want to solve for. In our case, we have a = 2, b = -9, and c = 7.

The solutions to a quadratic equation are also known as its roots or zeros. These are the values of 'x' that make the equation true. A quadratic equation can have two, one, or no real roots. Our mission is to find those roots for the equation 2x^2 - 9x + 7 = 0.

When we talk about solving quadratic equations, we're essentially trying to find the values of x that make the equation equal to zero. These values are also known as the roots or solutions of the equation. There are several methods to tackle this, and we'll cover the most common ones.

Methods to Solve Quadratic Equations

There are several methods we can use to solve quadratic equations, including:

  1. Factoring
  2. Completing the Square
  3. Quadratic Formula

We'll walk through each of these methods, showing you how they work and when they might be most useful. Each method has its strengths, and understanding them all will give you a well-rounded approach to solving quadratic equations.

1. Factoring the Quadratic Equation

Factoring is often the quickest and easiest method when it works. The idea behind factoring is to rewrite the quadratic equation as a product of two binomials. If we can factor the equation into the form (px + q)(rx + s) = 0, then we know that either (px + q) = 0 or (rx + s) = 0, which allows us to solve for x.

Step-by-step factoring for 2x^2 - 9x + 7 = 0:

  1. Look for two numbers that multiply to ac (2 * 7 = 14) and add up to b (-9). In this case, the numbers are -2 and -7 because (-2) * (-7) = 14 and (-2) + (-7) = -9.
  2. Rewrite the middle term (-9x) using these two numbers: 2x^2 - 2x - 7x + 7 = 0
  3. Factor by grouping:
    • Group the first two terms and the last two terms: (2x^2 - 2x) + (-7x + 7) = 0
    • Factor out the greatest common factor (GCF) from each group: 2x(x - 1) - 7(x - 1) = 0
  4. Notice that (x - 1) is a common factor. Factor it out: (2x - 7)(x - 1) = 0
  5. Set each factor equal to zero and solve for x:
    • 2x - 7 = 0 => 2x = 7 => x = 7/2
    • x - 1 = 0 => x = 1

So, the solutions are x = 7/2 and x = 1. Factoring is super efficient when you can easily find the right factors, making it a go-to method for many.

2. Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations, meaning it works for any quadratic equation, regardless of whether it can be factored easily. It’s derived from the process of completing the square and provides a direct way to find the solutions.

The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

Where 'a', 'b', and 'c' are the coefficients from the quadratic equation ax^2 + bx + c = 0.

Applying the quadratic formula to 2x^2 - 9x + 7 = 0:

  1. Identify a, b, and c:

    • a = 2
    • b = -9
    • c = 7

  2. Plug the values into the quadratic formula:

    x = (-(-9) ± √((-9)^2 - 4 * 2 * 7)) / (2 * 2)

  3. Simplify:

    x = (9 ± √(81 - 56)) / 4 x = (9 ± √25) / 4 x = (9 ± 5) / 4

  4. Solve for the two possible values of x:

    • x = (9 + 5) / 4 = 14 / 4 = 7/2
    • x = (9 - 5) / 4 = 4 / 4 = 1

As you can see, we get the same solutions as when we factored: x = 7/2 and x = 1. The quadratic formula is a lifesaver when factoring seems too tricky or impossible. It's a bit more involved in terms of calculation, but it's a reliable method to have in your toolkit.

3. Completing the Square

Completing the square is another method for solving quadratic equations. While it might seem a bit more complex than factoring or using the quadratic formula, it's a powerful technique that can also be used to derive the quadratic formula itself. It involves manipulating the equation to form a perfect square trinomial.

Step-by-step completing the square for 2x^2 - 9x + 7 = 0:

  1. Divide the entire equation by the coefficient of x^2 (which is 2) to make the leading coefficient 1: x^2 - (9/2)x + 7/2 = 0
  2. Move the constant term (7/2) to the right side of the equation: x^2 - (9/2)x = -7/2
  3. Take half of the coefficient of the x term (-9/2), square it, and add it to both sides of the equation. Half of -9/2 is -9/4, and squaring it gives us 81/16: x^2 - (9/2)x + 81/16 = -7/2 + 81/16
  4. Rewrite the left side as a perfect square and simplify the right side: (x - 9/4)^2 = (-56 + 81) / 16 (x - 9/4)^2 = 25 / 16
  5. Take the square root of both sides: x - 9/4 = ±√(25/16) x - 9/4 = ±5/4
  6. Solve for x:
    • x = 9/4 + 5/4 = 14/4 = 7/2
    • x = 9/4 - 5/4 = 4/4 = 1

Again, we arrive at the same solutions: x = 7/2 and x = 1. Completing the square is particularly useful in situations where you need to rewrite the quadratic equation in vertex form or when dealing with more complex problems in algebra and calculus.

Choosing the Best Method

So, with all these methods, how do you choose the best one for a particular equation? Here's a quick guide:

  • Factoring: If you can quickly identify factors, this is the fastest method. Always check if factoring is an option first.
  • Quadratic Formula: Use this when factoring is difficult or impossible. It's a foolproof method that always works.
  • Completing the Square: This is useful when you need to rewrite the equation in vertex form or when dealing with more theoretical problems. It’s also the method used to derive the quadratic formula.

Conclusion

Alright, guys! We've successfully solved the quadratic equation 2x^2 - 9x + 7 = 0 using three different methods: factoring, the quadratic formula, and completing the square. We found that the solutions are x = 7/2 and x = 1. Remember, each method has its strengths, and the best one to use depends on the specific equation you're dealing with. Keep practicing, and you'll become a quadratic equation-solving pro in no time!

Understanding these methods not only helps you solve equations but also provides a deeper insight into the nature of quadratic equations and their applications in various fields of mathematics and beyond. Whether you're tackling algebra problems, physics calculations, or engineering challenges, knowing how to solve quadratic equations is a valuable skill. So, keep honing your skills, and don't hesitate to explore more complex problems. Happy solving!