Solving The Quadratic Equation 7 + 44z = 35z² Finding Real Solutions
Hey everyone! Let's dive into the world of quadratic equations and learn how to find those real solutions. Quadratic equations are a fundamental part of algebra, and mastering them opens doors to more advanced math topics. In this article, we'll break down the process step by step, using the example equation 7 + 44z = 35z² to guide us. So, grab your calculators, and let's get started!
Understanding Quadratic Equations
Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is:
ax² + bx + c = 0
Where a, b, and c are constants, and x is the variable. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that make the equation true. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph.
Before we jump into solving, it's important to understand why quadratic equations are so important. You guys will see them everywhere, from physics problems describing projectile motion to engineering calculations for designing structures. They even pop up in economics and computer science! So, getting a solid handle on these equations is a smart move for your future studies and career.
Standard Form: The Key to Solving
The first step in solving any quadratic equation is to rewrite it in standard form: ax² + bx + c = 0. This form helps us easily identify the coefficients a, b, and c, which are crucial for applying solution methods like factoring, completing the square, or using the quadratic formula.
Why is the standard form so important? Well, think of it like having a recipe. The standard form lays out all the ingredients (the coefficients) in the right order so you can follow the steps correctly. Without it, you'd be trying to bake a cake with the ingredients all mixed up and the instructions in a jumble. Trust me, you want the recipe – I mean, the equation – in the standard form!
Our Example: 7 + 44z = 35z²
Let's take our example equation, 7 + 44z = 35z², and rearrange it into standard form. To do this, we need to move all terms to one side of the equation, leaving zero on the other side. Subtracting 7 and 44z from both sides, we get:
0 = 35z² - 44z - 7
Now, we can rewrite this as:
35z² - 44z - 7 = 0
See how we did that? We just shuffled things around to get that nice, clean standard form. Now we can easily see that a = 35, b = -44, and c = -7. These values are our keys to unlocking the solutions! It's like having the combination to a safe – once you have it, you can get to the treasure inside (which, in this case, is the solutions to our equation!).
Methods for Solving Quadratic Equations
There are several methods we can use to solve quadratic equations, each with its strengths and weaknesses. Let's explore three common methods: factoring, the quadratic formula, and completing the square. We'll focus on the quadratic formula for our example, but it's good to know the other options too.
Method 1 Factoring
Factoring involves expressing the quadratic equation as a product of two binomials. This method is efficient when the equation can be factored easily. For example, the equation x² - 5x + 6 = 0 can be factored as (x - 2)(x - 3) = 0. Setting each factor to zero gives us the solutions x = 2 and x = 3.
Factoring is like solving a puzzle – you're trying to find the two pieces that fit together perfectly to make the original equation. When it works, it's super satisfying and quick. But, like a puzzle, it's not always easy to see the solution right away, and some equations just can't be factored using simple methods.
Method 2: The Quadratic Formula
The quadratic formula is a universal method that works for any quadratic equation, regardless of whether it can be factored easily. The formula is:
x = (-b ± √(b² - 4ac)) / (2a)
This formula might look a bit intimidating at first, but it's a powerful tool once you get the hang of it. It basically takes the coefficients a, b, and c from the standard form of your equation and spits out the solutions. Think of it like a magic box – you feed in the numbers, and it gives you the answers!
Method 3: Completing the Square
Completing the square involves manipulating the quadratic equation to form a perfect square trinomial on one side. This method is useful for deriving the quadratic formula and can be applied even when factoring is difficult. However, it can be more complex and time-consuming than other methods.
Completing the square is like rearranging a messy room into a perfectly organized space. You're taking the equation and transforming it into a more manageable form. While it can be a bit more work, it's a powerful technique that can be really helpful in certain situations.
Applying the Quadratic Formula to Our Example
For our example equation, 35z² - 44z - 7 = 0, we've already identified the coefficients: a = 35, b = -44, and c = -7. Now, let's plug these values into the quadratic formula:
z = (-(-44) ± √((-44)² - 4 * 35 * -7)) / (2 * 35)
Let's break this down step by step to make sure we don't miss anything. It's like following a recipe – you want to add each ingredient in the right amount and at the right time to get the best result.
Step 1 Simplify the Expression
First, let's simplify the expression inside the square root and the rest of the equation:
z = (44 ± √(1936 + 980)) / 70
z = (44 ± √2916) / 70
Step 2: Calculate the Square Root
Next, we calculate the square root of 2916, which is 54:
z = (44 ± 54) / 70
Step 3: Find the Two Solutions
Now, we have two possible solutions, one with addition and one with subtraction:
- Solution 1:
z₁ = (44 + 54) / 70 = 98 / 70 = 7 / 5
- Solution 2:
z₂ = (44 - 54) / 70 = -10 / 70 = -1 / 7
So, there you have it! Our two real solutions for the quadratic equation 35z² - 44z - 7 = 0 are z = 7/5 and z = -1/7.
Verifying the Solutions
It's always a good idea to verify your solutions to ensure they are correct. We can do this by plugging each solution back into the original equation and checking if it holds true. This is like proofreading your work – you want to make sure you haven't made any mistakes along the way.
Verification for z = 7/5
Let's substitute z = 7/5 into the original equation:
7 + 44(7/5) = 35(7/5)²
7 + 308/5 = 35(49/25)
35/5 + 308/5 = 1715/25
343/5 = 343/5
The equation holds true, so z = 7/5 is indeed a solution.
Verification for z = -1/7
Now, let's substitute z = -1/7 into the original equation:
7 + 44(-1/7) = 35(-1/7)²
7 - 44/7 = 35(1/49)
49/7 - 44/7 = 35/49
5/7 = 5/7
This equation also holds true, confirming that z = -1/7 is a solution.
Conclusion
In this article, we've walked through the process of finding the real solutions of the quadratic equation 7 + 44z = 35z². We learned how to rewrite the equation in standard form, explored different solution methods, and applied the quadratic formula to find our solutions: z = 7/5 and z = -1/7. We also emphasized the importance of verifying solutions to ensure accuracy.
Mastering quadratic equations is a crucial step in your mathematical journey. By understanding the concepts and practicing the methods, you'll be well-equipped to tackle more complex problems in algebra and beyond. So keep practicing, guys, and you'll become quadratic equation whizzes in no time!
Now, armed with this knowledge, you can confidently approach quadratic equations and find their real solutions. Happy solving!
