Solving 3x^2 - 7x - 5 = 0: A Quadratic Formula Guide
Have you ever encountered a quadratic equation that just seems impossible to factor? Don't worry, you're not alone! That's where the quadratic formula comes in handy. It's a powerful tool that allows you to find the solutions (also called roots or zeros) of any quadratic equation, no matter how complex it looks. In this article, we'll break down how to use the quadratic formula step-by-step, using the equation 3x² - 7x - 5 = 0 as our example. We'll make it super easy to understand, so you can confidently tackle any quadratic equation that comes your way.
Understanding the Quadratic Formula
The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning it has the general form of ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The quadratic formula provides a way to solve for the values of x that satisfy this equation. It's a versatile tool in algebra and calculus, useful for solving a variety of problems where quadratic equations arise.
The quadratic formula itself looks like this:
x = [-b ± √(b² - 4ac)] / 2a
It might seem intimidating at first glance, but we'll break it down piece by piece. The symbols ± means that there are two possible solutions, one where you add the square root part and one where you subtract it. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant tells us about the nature of the solutions: if it's positive, there are two real solutions; if it's zero, there's one real solution (a repeated root); and if it's negative, there are two complex solutions. Understanding these components is the first step in mastering the formula.
Before we dive into solving our specific equation, let's quickly recap why this formula is so important. Factoring, another method for solving quadratic equations, isn't always straightforward or even possible for every equation. The quadratic formula, on the other hand, always works. It's your trusty backup plan when factoring fails, and it's an essential tool for anyone studying algebra or related fields. This formula ensures that you can consistently find the solutions to quadratic equations, regardless of their complexity. Now, let's put this knowledge into action and see how it works with our example equation.
Step 1: Identifying a, b, and c
The first step in using the quadratic formula is to correctly identify the coefficients a, b, and c from your quadratic equation. Remember, the standard form of a quadratic equation is ax² + bx + c = 0. So, we need to match the terms in our equation, 3x² - 7x - 5 = 0, with this standard form. This is a crucial step because using the wrong values will lead to incorrect solutions. Think of it as the foundation of your calculation – get it right, and the rest will follow smoothly.
In our equation, 3x² - 7x - 5 = 0, the coefficient of the x² term is a, the coefficient of the x term is b, and the constant term is c. By carefully comparing our equation to the standard form, we can see that:
- a = 3
- b = -7
- c = -5
It's important to pay attention to the signs! Notice that b is -7 and c is -5. These negative signs are crucial and must be included when you plug the values into the quadratic formula. A common mistake is forgetting the negative signs, which can completely change the outcome. Double-checking these values before moving on can save you a lot of trouble later. Once you've correctly identified a, b, and c, you're ready to move on to the next step: plugging these values into the formula. This is where the real magic happens, and we start to see the solutions unfold.
Step 2: Plugging the Values into the Formula
Now that we've identified a, b, and c, the next step is to carefully substitute these values into the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. This is where precision is key. It's like following a recipe – you need to add the right ingredients in the right amounts to get the desired result. A small mistake in substitution can throw off the entire calculation, so take your time and double-check your work.
Let's plug in our values (a = 3, b = -7, c = -5) into the formula:
x = [-(-7) ± √((-7)² - 4 * 3 * -5)] / (2 * 3)
Notice how we've replaced each variable with its corresponding value. It's especially important to use parentheses when substituting negative numbers, like we did with b. This helps to avoid sign errors. For instance, -(-7) becomes positive 7, while (-7)² becomes positive 49. These small details can make a big difference in the final answer. This step is a direct application of the formula, turning abstract symbols into concrete numbers. With the values substituted, we're now ready to simplify the expression and move closer to finding our solutions. The next step involves carefully performing the arithmetic operations to unravel the equation further.
Step 3: Simplifying the Expression
After plugging the values into the quadratic formula, the next step is to simplify the expression. This involves carefully performing the arithmetic operations in the correct order (remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Simplifying step-by-step not only makes the calculation more manageable but also reduces the chances of making errors. It's like breaking down a large task into smaller, more achievable steps.
Let's continue from where we left off:
x = [-(-7) ± √((-7)² - 4 * 3 * -5)] / (2 * 3)
First, let's simplify the terms inside the square root and the denominator:
x = [7 ± √(49 + 60)] / 6
We've simplified -(-7) to 7, (-7)² to 49, and -4 * 3 * -5 to 60. The denominator 2 * 3 is simplified to 6. Now, let's add the numbers inside the square root:
x = [7 ± √109] / 6
At this point, we've simplified the expression as much as possible without approximating the square root of 109. Since 109 is not a perfect square, √109 is an irrational number. This means it cannot be expressed as a simple fraction, and its decimal representation goes on infinitely without repeating. This is perfectly normal, and it simply means our solutions will involve the square root of 109. We're now ready for the final step, where we'll separate the ± into two separate solutions and, if needed, approximate the square root to get decimal answers.
Step 4: Finding the Two Solutions
The ± symbol in the quadratic formula is a shorthand way of representing two different solutions. It indicates that we need to perform the calculation twice: once with addition and once with subtraction. This is because quadratic equations generally have two solutions (although sometimes they can be the same). Separating these two solutions is the final step in solving the equation.
From our simplified expression, x = [7 ± √109] / 6, we can now write out the two solutions:
- x₁ = (7 + √109) / 6
- x₂ = (7 - √109) / 6
These are the exact solutions to the equation 3x² - 7x - 5 = 0. Depending on the context of the problem, you might need to approximate these solutions as decimal numbers. To do this, you would use a calculator to find the square root of 109 (which is approximately 10.44) and then perform the arithmetic.
Let's approximate the solutions:
- x₁ ≈ (7 + 10.44) / 6 ≈ 17.44 / 6 ≈ 2.91
- x₂ ≈ (7 - 10.44) / 6 ≈ -3.44 / 6 ≈ -0.57
So, the approximate solutions to the equation 3x² - 7x - 5 = 0 are x ≈ 2.91 and x ≈ -0.57. These are the values of x that, when plugged back into the original equation, would make the equation true (or very close to true, given the approximation). This completes our journey of solving the quadratic equation using the quadratic formula. We've gone from identifying the coefficients to plugging them into the formula, simplifying, and finally, finding the two solutions. The quadratic formula is a powerful tool, and with practice, you'll become more and more comfortable using it to solve a wide range of quadratic equations.
Conclusion
In this guide, we've walked through the process of using the quadratic formula to solve the equation 3x² - 7x - 5 = 0. We've broken down each step, from identifying the coefficients to simplifying the expression and finding the two solutions. The quadratic formula is a versatile tool that can be used to solve any quadratic equation, and with practice, you'll become more confident in your ability to use it.
Remember, the key to mastering the quadratic formula is to understand each step and practice consistently. Start by identifying a, b, and c correctly, then carefully substitute these values into the formula. Simplify the expression step-by-step, and don't forget to consider both the positive and negative roots when finding the two solutions. With these skills, you'll be well-equipped to tackle any quadratic equation that comes your way. For more in-depth explanations and examples, you can visit Khan Academy's Quadratic Formula page.
