Hey guys! Today, we're diving into a quadratic equation problem that might seem a bit daunting at first, but trust me, we'll break it down step by step and you'll see it's totally manageable. Our mission is to figure out which value is a solution for the equation (7/4)x^2 - 2 = -0.5x + 4. We've got four options to choose from: A) x = -2, B) x = -1, C) x = 1, and D) x = 5. So, let's roll up our sleeves and get to work!
Understanding the Problem: Quadratic Equations Explained
Before we jump into solving, let's quickly recap what a quadratic equation is. In simple terms, it's an equation that can be written in the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to find. The key characteristic of a quadratic equation is the presence of the x^2 term, which makes it a second-degree polynomial. This means it can have up to two solutions, also known as roots or zeros.
Our equation, (7/4)x^2 - 2 = -0.5x + 4, fits this description perfectly. The goal is to find the value(s) of 'x' that make the equation true. There are several methods to solve quadratic equations, such as factoring, completing the square, using the quadratic formula, or, in our case, testing the given options. Since we have specific values to check, the most straightforward approach is to substitute each value into the equation and see if it holds true. This method is especially handy when you have multiple-choice options, like we do.
The Importance of Solutions in Quadratic Equations
Why are we even bothering to find these solutions? Well, the solutions of a quadratic equation have significant practical applications. Imagine you're designing a bridge or calculating the trajectory of a projectile. Quadratic equations can help you model these real-world scenarios accurately. The solutions represent the points where the parabola (the graph of a quadratic equation) intersects the x-axis. Understanding these points can provide valuable insights into the behavior of the system you're modeling.
Moreover, quadratic equations pop up in various fields like physics, engineering, economics, and computer science. They're fundamental tools for describing and predicting phenomena that involve curves or parabolic paths. So, mastering the art of solving quadratic equations is a crucial skill in many disciplines. Knowing how to find the correct solutions ensures that you can make accurate predictions and informed decisions in these fields.
Method 1: Testing the Solutions - A Step-by-Step Guide
The most direct way to tackle this problem is by testing each of the provided solutions. We'll substitute each value of 'x' into the equation (7/4)x^2 - 2 = -0.5x + 4 and see if the left side equals the right side. If it does, we've found a solution! Let's start with option A, x = -2.
Step 1: Substitute x = -2
Replace 'x' with -2 in the equation: (7/4)(-2)^2 - 2 = -0.5(-2) + 4. Now, we need to simplify both sides of the equation separately.
Step 2: Simplify the Left Side
First, calculate (-2)^2, which equals 4. Then, multiply by 7/4: (7/4) * 4 = 7. Subtract 2 from the result: 7 - 2 = 5. So, the left side simplifies to 5.
Step 3: Simplify the Right Side
Multiply -0.5 by -2, which gives us 1. Add 4 to the result: 1 + 4 = 5. The right side also simplifies to 5.
Step 4: Compare Both Sides
We found that both the left side and the right side of the equation are equal to 5 when x = -2. This means x = -2 is a solution to the equation. Hooray!
Repeating the Process for Other Options
Even though we've found a solution, it's good practice to check the other options to ensure there isn't another correct answer or to simply reinforce our understanding. Let's quickly run through options B, C, and D.
For x = -1: (7/4)(-1)^2 - 2 = (7/4) - 2 = -1/4 on the left side, and -0.5(-1) + 4 = 0.5 + 4 = 4.5 on the right side. These are not equal.
For x = 1: (7/4)(1)^2 - 2 = (7/4) - 2 = -1/4 on the left side, and -0.5(1) + 4 = -0.5 + 4 = 3.5 on the right side. Again, these are not equal.
For x = 5: (7/4)(5)^2 - 2 = (7/4)(25) - 2 = 43.75 - 2 = 41.75 on the left side, and -0.5(5) + 4 = -2.5 + 4 = 1.5 on the right side. Clearly, these are not equal either.
By testing all the options, we've confirmed that x = -2 is the only solution among the choices provided.
Method 2: Rearranging and Solving the Quadratic Equation
Now, let's explore another approach to solve this equation: rearranging it into the standard quadratic form and using the quadratic formula. This method is a bit more involved but provides a deeper understanding of the equation's structure.
Step 1: Rearrange the Equation
Our original equation is (7/4)x^2 - 2 = -0.5x + 4. To get it into the standard form (ax^2 + bx + c = 0), we need to move all terms to one side. Let's add 0.5x to both sides and subtract 4 from both sides:
(7/4)x^2 + 0.5x - 2 - 4 = 0
This simplifies to:
(7/4)x^2 + 0.5x - 6 = 0
Step 2: Eliminate the Fraction (Optional but Recommended)
To make things easier, let's get rid of the fraction by multiplying the entire equation by 4:
4 * [(7/4)x^2 + 0.5x - 6] = 4 * 0
This gives us:
7x^2 + 2x - 24 = 0
Now we have a standard quadratic equation without fractions.
Step 3: Apply the Quadratic Formula
The quadratic formula is a powerful tool for solving any quadratic equation in the form ax^2 + bx + c = 0. It states that:
x = [-b ± √(b^2 - 4ac)] / (2a)
In our equation, a = 7, b = 2, and c = -24. Let's plug these values into the formula:
x = [-2 ± √(2^2 - 4 * 7 * -24)] / (2 * 7)
Step 4: Simplify the Formula
Let's break down the simplification:
First, calculate the discriminant (the part under the square root): 2^2 - 4 * 7 * -24 = 4 + 672 = 676
Now, find the square root of 676, which is 26.
So, the formula becomes:
x = [-2 ± 26] / 14
This gives us two possible solutions:
x1 = (-2 + 26) / 14 = 24 / 14 = 12 / 7
x2 = (-2 - 26) / 14 = -28 / 14 = -2
Step 5: Compare with the Options
We found two solutions: x = 12/7 and x = -2. Looking back at our options, we see that x = -2 is indeed one of the choices. The other solution, x = 12/7, is not among the given options, but it's still a valid solution to the equation.
Why Both Methods Matter
As you can see, we arrived at the same answer using two different methods. Testing the solutions is quicker when you have specific values to check, while rearranging and using the quadratic formula provides a more general approach that works for any quadratic equation. Understanding both methods gives you flexibility and a deeper understanding of quadratic equations.
When to Use Each Method
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Testing Solutions: This method is ideal when you have multiple-choice questions or a limited set of possible solutions to check. It's straightforward and often faster than other methods.
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Rearranging and Using the Quadratic Formula: This method is more suitable when you need to find all possible solutions to the equation, especially when the solutions aren't obvious or easily guessed. It's a reliable method that works for any quadratic equation.
Conclusion: The Solution is x = -2
After exploring two different methods, we've confidently determined that the value x = -2 is a solution of the equation (7/4)x^2 - 2 = -0.5x + 4. We achieved this by both substituting the given values and rearranging the equation to apply the quadratic formula. Both approaches are valuable tools in your mathematical toolkit.
Key Takeaways
- Quadratic equations can be solved using various methods, including testing solutions and applying the quadratic formula.
- Testing solutions is efficient when you have specific values to check.
- Rearranging and using the quadratic formula provides a general approach for solving any quadratic equation.
- Understanding both methods enhances your problem-solving skills.
So, there you have it, guys! We've successfully navigated this quadratic equation and found our solution. Remember, practice makes perfect, so keep honing your skills, and you'll become a quadratic equation pro in no time!