Solve -7x^2 + 4x + 1 = 0: Quadratic Equation Solutions

by ADMIN 55 views

Hey everyone! Let's dive into the world of quadratic equations. Today, we're going to tackle the equation $-7x^2 + 4x + 1 = 0$. We'll explore the different methods to solve it and break down the solution step by step. So, if you've ever felt lost when faced with a quadratic equation, you're in the right place! Let's get started and make these equations less intimidating.

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means it has the general form: $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. The solutions to the quadratic equation, also known as the roots or zeros, are the values of x that satisfy the equation. These roots represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis.

The Importance of Identifying Coefficients

The first crucial step in solving a quadratic equation is to correctly identify the coefficients a, b, and c. These coefficients play a vital role in determining the solutions. In our equation, $-7x^2 + 4x + 1 = 0$, we have:

  • a = -7
  • b = 4
  • c = 1

Getting these right is key because they'll be plugged into our chosen solution method. A slight mistake here can throw off your entire answer, so always double-check! Understanding these coefficients not only helps in solving the equation but also in visualizing the parabola represented by the equation. The coefficient a tells us about the parabola's direction (whether it opens upwards or downwards) and its 'width', while b and c influence its position on the coordinate plane. It's like having the secret code to unlock the equation's mysteries!

Methods to Solve Quadratic Equations

There are several methods to solve quadratic equations, each with its own strengths and when to use it:

  1. Factoring: This method involves breaking down the quadratic expression into two linear expressions. It's quick and easy when it works, but it's not always applicable. Factoring is like finding the perfect puzzle pieces that fit together to form the quadratic equation. When you can spot those pieces, the solution falls into place almost effortlessly. For example, if you can rewrite the equation in the form (x + p)(x + q) = 0, then the solutions are simply x = -p and x = -q. However, not all quadratic equations are easily factorable, which is where other methods come in handy.

  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial. It's a bit more involved but always works. Completing the square is like transforming the equation into a shape that's easier to handle. It involves adding and subtracting a specific value to create a perfect square, which then allows you to solve for x by taking the square root. This method is particularly useful when the equation is not easily factorable, and it also provides a deeper understanding of the structure of quadratic equations.

  3. Quadratic Formula: This formula provides a direct solution for any quadratic equation. It's a reliable method that always works, making it a favorite among students and mathematicians alike. The quadratic formula is the ultimate tool in your quadratic equation-solving arsenal. It's like having a universal key that unlocks the solution to any quadratic equation, no matter how complex. By simply plugging in the coefficients a, b, and c into the formula, you can find the solutions without having to factor or complete the square. It's a powerful and versatile method that ensures you can always find the answer.

Why the Quadratic Formula is a Go-To Method

While factoring can be quicker sometimes, the quadratic formula is the most reliable because it works for any quadratic equation, no matter how messy the coefficients are. Factoring is like finding the right key to open a lock – sometimes you get it on the first try, but other times you have to try many keys. The quadratic formula, on the other hand, is like having a master key that unlocks every quadratic equation. It's a surefire method that doesn't depend on clever guesswork or spotting patterns. This makes it an invaluable tool, especially in exams or situations where accuracy and speed are crucial. The formula itself might look a bit intimidating at first, but once you've used it a few times, it becomes second nature.

Applying the Quadratic Formula

Given the equation $-7x^2 + 4x + 1 = 0$, we've already identified our coefficients:

  • a = -7
  • b = 4
  • c = 1

Now, let's plug these values into the quadratic formula:

x=−bext±extb2−4ac2ax = \frac{-b ext{±} ext{\sqrt{b^2 - 4ac}}}{2a}

Substituting the values, we get:

x=−4ext±ext42−4(−7)(1)2(−7)x = \frac{-4 ext{±} ext{\sqrt{4^2 - 4(-7)(1)}}}{2(-7)}

Step-by-Step Breakdown of the Substitution

Let's walk through this substitution step by step to make sure we're crystal clear. The quadratic formula might seem intimidating at first, but breaking it down into smaller parts makes it much more manageable. We're essentially replacing the letters in the formula with the numbers from our equation. First, we see "-b" in the formula. Since our b value is 4, "-b" becomes -4. Next, we have the square root part. Inside the square root, we have "b²", which is 4² or 16. Then, we have "-4ac". Our a is -7 and our c is 1, so "-4ac" becomes -4 * -7 * 1, which equals 28. Finally, in the denominator, we have "2a", which is 2 * -7, or -14. Putting it all together, we get a much clearer picture of how the values fit into the formula. This meticulous approach helps prevent errors and builds confidence in using the quadratic formula.

Simplifying the Expression

Now, let's simplify the expression we've got. This is where we put on our algebraic detective hats and start unraveling the equation. We've got a few layers to peel back, starting with the square root. Inside the square root, we need to calculate the value of 4² - 4(-7)(1). Remember the order of operations (PEMDAS/BODMAS): we do the multiplication first. So, 4² is 16, and -4 * -7 * 1 is 28. Now we add those together: 16 + 28 gives us 44. So, the square root becomes √44. Next, we look at the denominator, which is 2 * -7, and that simplifies to -14. Now our equation looks a bit cleaner, but we can still simplify further. The square root of 44 can be simplified because 44 has a perfect square factor. We can rewrite √44 as √(4 * 11), and since √4 is 2, we can simplify it to 2√11. This step is crucial because it helps us express the solution in its simplest form, which is often required in exams and problem-solving. By breaking down the simplification process into these small steps, we make it less daunting and ensure we don't miss any opportunities to make the equation easier to handle.

x=−4ext±ext16+28−14x = \frac{-4 ext{±} ext{\sqrt{16 + 28}}}{-14}

x=−4ext±ext44−14x = \frac{-4 ext{±} ext{\sqrt{44}}}{-14}

We can simplify $ ext{\sqrt{44}}$ as $ ext{\sqrt{4 imes 11}} = 2 ext{\sqrt{11}}$. So our equation becomes:

x=−4ext±2ext11−14x = \frac{-4 ext{±} 2 ext{\sqrt{11}}}{-14}

Final Simplification and the Two Solutions

We're almost there, guys! Just a bit more simplification to do. Notice that we can divide both the numerator and the denominator by -2. This is a neat trick that helps us clean up the fraction and get it into its simplest form. When we divide -4 by -2, we get 2. When we divide 2√11 by -2, we get -√11. And when we divide -14 by -2, we get 7. So, our equation transforms into: x = (2 ± √11) / 7. Now, let's talk about what that "±" symbol means. It's like a shortcut that represents two separate solutions. One solution is when we use the plus sign, and the other is when we use the minus sign. This is because quadratic equations, due to their squared term, often have two possible solutions. So, we have two solutions: x = (2 + √11) / 7 and x = (2 - √11) / 7. These are the points where the parabola crosses the x-axis. Each solution is a valid answer, and they both satisfy the original equation. Understanding this dual nature of solutions is key to mastering quadratic equations. It's not just about finding one answer; it's about recognizing the two possibilities that arise from the equation's structure. This final simplification step brings clarity to our solutions and highlights the elegance of the quadratic formula.

x=2ext±ext117x = \frac{2 ext{±} ext{\sqrt{11}}}{7}

Therefore, the correct answer is: $ rac{2 ext{±} ext{\sqrt{11}}}{7}$

Common Mistakes to Avoid

When working with quadratic equations, there are a few common pitfalls that students often stumble into. Let's highlight these so you can steer clear of them. One frequent error is with the signs. It's super important to keep track of negative signs, especially when substituting values into the quadratic formula or simplifying expressions. A misplaced negative sign can completely change your answer. Another mistake is messing up the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. Skipping a step or doing them in the wrong order can lead to incorrect results. Also, people sometimes forget to simplify the square root. Always look for perfect square factors within the square root and simplify them as much as possible. Lastly, don't forget that quadratic equations usually have two solutions. Make sure you account for both the plus and minus options when using the quadratic formula. By being mindful of these common errors, you can increase your accuracy and confidence in solving quadratic equations. It's all about paying attention to the details and double-checking your work.

Conclusion

Great job, guys! We've successfully solved the quadratic equation $-7x^2 + 4x + 1 = 0$ using the quadratic formula. Remember, the key to mastering these equations is understanding the basics, knowing your formulas, and practicing consistently. Keep up the great work, and you'll be a quadratic equation pro in no time! Solving quadratic equations might seem like a daunting task at first, but with the right tools and a bit of practice, you can conquer them with confidence. The quadratic formula is your best friend in this journey, providing a reliable method to find the solutions. Remember to break down the problem into manageable steps, double-check your work, and don't be afraid to ask for help when you need it. The world of mathematics is full of exciting challenges, and with each equation you solve, you're building your skills and expanding your understanding. So, keep exploring, keep learning, and most importantly, keep having fun with math!