Unlocking Solutions: Using The Quadratic Formula

by Alex Johnson 49 views

Hey math enthusiasts! Let's dive into the fascinating world of algebra and tackle a classic problem: solving a quadratic equation. Specifically, we're going to use the quadratic formula to find the values of x in the equation x² = 5 - x. This formula is a powerful tool, a mathematical Swiss Army knife, if you will, that allows us to find the solutions (also known as roots) of any quadratic equation. Don't worry if the term “quadratic” sounds intimidating; we'll break it down step-by-step to make it crystal clear. So, grab your pencils (or your favorite note-taking app), and let's get started!

Understanding Quadratic Equations

First, let's talk about what a quadratic equation actually is. In simple terms, a quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The “quadratic” part comes from the fact that the highest power of the variable (in this case, x) is 2. The a, b, and c are just numbers. For instance, in the equation 2x² + 3x - 5 = 0, a would be 2, b would be 3, and c would be -5. The solutions to a quadratic equation represent the points where the corresponding parabola (the U-shaped curve you get when you graph a quadratic equation) crosses the x-axis. These points are also known as the roots or zeros of the equation. Understanding this structure is crucial because the quadratic formula is designed specifically to solve equations in this format. Rearranging our original equation, x² = 5 - x, to match this standard form is the first essential step. The goal is to set the equation to zero, preparing it for the formula.

Rearranging the Equation

Our initial equation, x² = 5 - x, isn't in the standard ax² + bx + c = 0 format yet. To get it there, we need to move all the terms to one side of the equation, leaving zero on the other side. This is like tidying up a room by moving all the clutter to one corner. Here's how we do it step-by-step:

  1. Add x to both sides: This eliminates the x term from the right side. This gives us x² + x = 5.
  2. Subtract 5 from both sides: This moves the constant term to the left side and sets the right side to zero. This results in x² + x - 5 = 0.

Now, our equation is in the standard quadratic form, which means we can easily identify the values of a, b, and c: a = 1, b = 1, and c = -5. Make sure you get the signs correct. This is really critical because a minus sign in the wrong place can completely change your solution!

The Quadratic Formula: Your Solution Blueprint

Now that our equation is formatted correctly, it's time to unleash the quadratic formula. This formula is the key to solving for x in any quadratic equation and it goes like this:

x = (-b ± √(b² - 4ac)) / 2a

Don't let the formula intimidate you! It looks complex, but it's just a recipe; you plug in your values for a, b, and c, and carefully follow the steps. The '±' symbol means that there are two possible solutions for x: one where you add the square root part, and one where you subtract it. So, let’s go ahead and carefully substitute the values of a, b, and c that we identified earlier (a = 1, b = 1, and c = -5) into the formula.

Plugging in the Values

Let’s plug the values into the formula and see what we get. So we're going to replace a, b, and c with their corresponding numerical values, like this:

x = (-1 ± √(1² - 4 * 1 * -5)) / (2 * 1)

See how we've replaced each variable with its corresponding number? Now, our equation is all numbers, and we are ready to carefully evaluate. Double-checking each step is useful here, because one small arithmetic mistake can cause big problems down the line.

Simplifying the Formula

Now we'll simplify this equation step-by-step. Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

  1. Simplify the expression inside the square root:

    • First, square 1: 1² = 1.
    • Next, multiply 4 by 1 and by -5: 4 * 1 * -5 = -20.
    • Finally, subtract -20 from 1: 1 - (-20) = 1 + 20 = 21. This means we have a positive number, so the solutions will be real numbers.

  2. Simplify the denominator: 2 * 1 = 2.

After these simplifications, our formula now looks like this: x = (-1 ± √21) / 2.

Finding the Solutions: The Final Step

We're in the home stretch! The formula has now been simplified to find the solutions. Now, we have two possible values for x, one where we add the square root of 21, and one where we subtract it. Let's find those two values.

Calculating the Two Possible Solutions

  1. Solution 1 (using the plus sign): x = (-1 + √21) / 2. We can use a calculator to find the approximate value of √21, which is roughly 4.58. So, x ≈ (-1 + 4.58) / 2 = 3.58 / 2 ≈ 1.79. This is one possible solution for x.
  2. Solution 2 (using the minus sign): x = (-1 - √21) / 2. Using our calculator again, we find that x ≈ (-1 - 4.58) / 2 = -5.58 / 2 ≈ -2.79. This is the second possible solution for x.

Therefore, using the quadratic formula, we have found that the solutions to the equation x² = 5 - x are approximately x ≈ 1.79 and x ≈ -2.79. These are the points where the parabola representing the equation x² + x - 5 = 0 crosses the x-axis. Using the quadratic formula guarantees that you find all the real number solutions to a quadratic equation.

Key Takeaways and Tips for Success

Let’s recap what we've learned and some useful tips to ensure you master the quadratic formula:

  • Standard Form is Key: Always rearrange your quadratic equation into the standard form ax² + bx + c = 0 before applying the formula.
  • Careful with Signs: Pay very close attention to the signs of a, b, and c. A small mistake here can completely change your answer.
  • Order of Operations: Follow the order of operations (PEMDAS/BODMAS) meticulously when simplifying the formula.
  • Check Your Work: Whenever possible, check your answers by plugging them back into the original equation. This is a great way to catch any errors.
  • Practice, Practice, Practice: The more you practice using the quadratic formula, the more comfortable and confident you’ll become. Try solving different quadratic equations to hone your skills.
  • Use a Calculator: Don’t hesitate to use a calculator to evaluate square roots and perform calculations. This saves time and helps reduce the risk of arithmetic errors.

Conclusion

And there you have it! We've successfully used the quadratic formula to solve for x in the equation x² = 5 - x. This formula is incredibly versatile, and knowing how to use it is a valuable skill in algebra and beyond. This is why it is so important and is part of the core math curriculum. By following the steps outlined above, you can confidently tackle any quadratic equation that comes your way. Keep practicing, stay curious, and you'll find that math, like everything, gets easier with practice!

I hope this step-by-step guide has been helpful! If you have any questions, feel free to ask. Happy solving!

If you want to read more about Quadratic Equation, check this link: Khan Academy