Solving 3x^2 + X - 2 = 0: A Step-by-Step Guide

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Hey guys! Ever get stuck on a quadratic equation and feel like you're staring at a jumbled mess of numbers and symbols? Well, you're not alone! Quadratic equations can seem intimidating, but once you break them down, they're actually pretty manageable. Today, we're going to tackle the equation 3x^2 + x - 2 = 0. We'll walk through the solution step-by-step, so you'll be a quadratic equation-solving pro in no time. Let's dive in!

Understanding Quadratic Equations

Before we jump into solving, let's make sure we're all on the same page about what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. That basically means the highest power of the variable (in this case, 'x') is 2. The general form of a quadratic equation is:

ax^2 + bx + c = 0

Where 'a', 'b', and 'c' are coefficients (just fancy words for numbers) and 'x' is the variable we're trying to solve for. In our equation, 3x^2 + x - 2 = 0, we can see that:

  • a = 3
  • b = 1 (because the coefficient in front of 'x' is 1)
  • c = -2

Now that we know what a quadratic equation is, let's talk about how to solve them. There are a few different methods we can use, but we're going to focus on factoring and the quadratic formula in this guide. Factoring is like reverse distribution – we're trying to find two binomials that multiply together to give us our quadratic equation. The quadratic formula is a trusty tool that works for any quadratic equation, even the ones that are hard to factor. We will primarily focus on the factoring method and touch on the quadratic formula as an alternative.

Method 1: Factoring the Quadratic Equation

Factoring is a fantastic technique when it works, because it can be the quickest way to find the solutions. It involves breaking down the quadratic expression into two binomials. Here’s how we can factor 3x^2 + x - 2 = 0:

1. Find two numbers that multiply to ac and add up to b.

Remember those coefficients we identified earlier? a = 3, b = 1, and c = -2. So:

  • ac = 3 * (-2) = -6
  • We need two numbers that multiply to -6 and add up to 1.

After a little thought, we can see that the numbers 3 and -2 fit the bill:

  • 3 * -2 = -6
  • 3 + (-2) = 1

2. Rewrite the middle term (bx) using these two numbers.

Instead of '+ x', we'll write '+ 3x - 2x'. This doesn't change the equation, it just splits the 'x' term into two parts:

3x^2 + 3x - 2x - 2 = 0

3. Factor by grouping.

Now, we'll group the first two terms and the last two terms together and factor out the greatest common factor (GCF) from each group:

  • From the first group (3x^2 + 3x), the GCF is 3x. Factoring this out gives us: 3x(x + 1)
  • From the second group (-2x - 2), the GCF is -2. Factoring this out gives us: -2(x + 1)

So, our equation now looks like this:

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

Notice that both terms now have a common factor of (x + 1). This is what we want!

4. Factor out the common binomial factor.

We can factor out (x + 1) from the entire equation:

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

5. Set each factor equal to zero and solve for x.

For the product of two factors to be zero, at least one of them must be zero. So, we set each factor equal to zero and solve:

  • x + 1 = 0 => x = -1
  • 3x - 2 = 0 => 3x = 2 => x = 2/3

So, our solutions are x = -1 and x = 2/3. We've successfully solved the quadratic equation by factoring!

Method 2: Using the Quadratic Formula

Okay, so factoring is awesome when it works, but sometimes, the equation is just too tricky to factor. That's where the quadratic formula comes in! It's a bit more involved, but it's a guaranteed way to find the solutions to any quadratic equation.

The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

Don't let it scare you! It's just a matter of plugging in the values of a, b, and c that we already identified.

Let's use it to solve our equation, 3x^2 + x - 2 = 0:

  • a = 3
  • b = 1
  • c = -2

1. Plug the values into the formula.

x = (-1 ± √(1^2 - 4 * 3 * -2)) / (2 * 3)

2. Simplify the expression.

x = (-1 ± √(1 + 24)) / 6 x = (-1 ± √25) / 6 x = (-1 ± 5) / 6

3. Calculate the two possible solutions.

Remember that ± sign? It means we have two possible solutions:

  • x = (-1 + 5) / 6 = 4 / 6 = 2/3
  • x = (-1 - 5) / 6 = -6 / 6 = -1

And guess what? We got the same solutions as we did with factoring: x = -1 and x = 2/3. The quadratic formula is a reliable way to double-check your work or solve equations that are tough to factor.

Why are these the solutions?

So, we've found that x = -1 and x = 2/3 are the solutions to the equation 3x^2 + x - 2 = 0. But what does that actually mean? The solutions to a quadratic equation are also called the roots or zeros of the equation. They're the values of 'x' that make the equation true – when you plug them back into the equation, the result is zero. Graphically, these solutions are the x-intercepts of the parabola represented by the quadratic equation. In other words, they're the points where the parabola crosses the x-axis.

To verify this, let's substitute our solutions back into the original equation:

For x = -1:

3(-1)^2 + (-1) - 2 = 3(1) - 1 - 2 = 3 - 1 - 2 = 0

For x = 2/3:

3(2/3)^2 + (2/3) - 2 = 3(4/9) + 2/3 - 2 = 4/3 + 2/3 - 2 = 6/3 - 2 = 2 - 2 = 0

As you can see, both solutions make the equation equal to zero, confirming that they are indeed the correct roots.

Conclusion

So there you have it! We've successfully solved the quadratic equation 3x^2 + x - 2 = 0 using both factoring and the quadratic formula. We found that the solutions are x = -1 and x = 2/3. Remember, quadratic equations might seem tricky at first, but with practice and the right tools, you can conquer them! Whether you prefer factoring or the quadratic formula, having these methods in your toolbox will make you a quadratic equation-solving master. Keep practicing, and you'll be amazed at what you can achieve! And hey, if you get stuck again, just remember this guide – we're here to help you out. Happy solving!