Solving 2x² + 2x - 6 = 0 A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're going to dive into solving a quadratic equation. Quadratic equations are those fascinating equations with a variable raised to the power of two (that little ² symbol). They pop up everywhere in math and real-world applications, so mastering them is super useful. Let's tackle the equation 2x² + 2x - 6 = 0 together.
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what quadratic equations are all about. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we want to find. The solutions for x are also known as the roots or zeros of the equation. These roots are the values of x that make the equation true. Think of it like finding the magic numbers that make the equation balance perfectly. There are several methods to solve these equations, but we'll focus on one of the most reliable methods: the quadratic formula.
In our equation, 2x² + 2x - 6 = 0, we can easily identify the constants: a = 2, b = 2, and c = -6. These values are crucial for using the quadratic formula. Recognizing these constants correctly is the first step towards successfully solving the equation. Now that we've got our constants, let's explore the powerful tool that will help us find the values of x.
The Mighty Quadratic Formula
The quadratic formula is a universal tool for solving quadratic equations. It's like a Swiss Army knife for math problems! The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Don't let the symbols scare you; it's actually quite straightforward once you get the hang of it. The ± symbol means we'll have two possible solutions, one with addition and one with subtraction. This accounts for the two roots that a quadratic equation can have. Now, let's plug in the values from our equation into the formula. Remember, a = 2, b = 2, and c = -6. Substituting these values, we get:
x = (-2 ± √(2² - 4 * 2 * -6)) / (2 * 2)
See how we just replaced a, b, and c with their respective values? Now, it's time to simplify this expression. We'll start by tackling the expression under the square root, which is called the discriminant. The discriminant tells us a lot about the nature of the roots, but more on that later. For now, let's just focus on simplifying it.
Step-by-Step Solution
Let's break down the solution step by step to make it crystal clear. First, we'll simplify the expression under the square root:
- Calculate 2²: 2² = 4
- Calculate 4 * 2 * -6: 4 * 2 * -6 = -48
- Subtract -48 from 4: 4 - (-48) = 4 + 48 = 52
So, our equation now looks like this:
x = (-2 ± √52) / (2 * 2)
Next, we simplify the denominator:
- 2 * 2 = 4
Now our equation looks even simpler:
x = (-2 ± √52) / 4
We're getting there! Now, let's simplify the square root of 52. We can break down 52 into its prime factors to simplify it. 52 can be written as 4 * 13, so √52 = √(4 * 13) = √4 * √13 = 2√13. Replacing √52 with 2√13, we get:
x = (-2 ± 2√13) / 4
We can simplify this further by dividing both the numerator and the denominator by 2:
x = (-1 ± √13) / 2
Finding the Two Roots
Remember that ± symbol? It means we have two solutions to consider. Let's separate them and find the two values of x.
Solution 1: Using the Plus Sign
For the first solution, we use the plus sign:
x₁ = (-1 + √13) / 2
√13 is approximately 3.60555. So, plugging that in:
x₁ = (-1 + 3.60555) / 2
x₁ = 2.60555 / 2
x₁ ≈ 1.30 (rounded to two decimal places)
Solution 2: Using the Minus Sign
For the second solution, we use the minus sign:
x₂ = (-1 - √13) / 2
Again, √13 is approximately 3.60555. So, plugging that in:
x₂ = (-1 - 3.60555) / 2
x₂ = -4.60555 / 2
x₂ ≈ -2.30 (rounded to two decimal places)
So, our two solutions are approximately x₁ ≈ 1.30 and x₂ ≈ -2.30. These are the values of x that make the equation 2x² + 2x - 6 = 0 true.
Verification
To be absolutely sure we've got the correct solutions, it's always a good idea to plug them back into the original equation. Let's verify our solutions:
Verification for x ≈ 1.30
Plug x = 1.30 into 2x² + 2x - 6 = 0:
2(1.30)² + 2(1.30) - 6
2(1.69) + 2.60 - 6
3.38 + 2.60 - 6
5.98 - 6
-0.02 ≈ 0
This is very close to zero, which means our solution is correct, considering we rounded to two decimal places.
Verification for x ≈ -2.30
Plug x = -2.30 into 2x² + 2x - 6 = 0:
2(-2.30)² + 2(-2.30) - 6
2(5.29) - 4.60 - 6
10.58 - 4.60 - 6
10.58 - 10.60
-0.02 ≈ 0
Again, this is very close to zero, confirming that our second solution is also correct.
Conclusion
Awesome job, guys! We've successfully solved the quadratic equation 2x² + 2x - 6 = 0 using the quadratic formula. We found two solutions: x ≈ 1.30 and x ≈ -2.30. Remember, quadratic equations can have two, one, or no real solutions, and the quadratic formula is a powerful tool to find them. Keep practicing, and you'll become a quadratic equation-solving pro in no time! If you have any questions or want to explore more math problems, feel free to ask. Happy solving!
