Solving 7/19x² + 7/19x = 0: A Step-by-Step Guide

by Rajiv Sharma 49 views

Hey guys! Today, we're diving deep into the fascinating world of quadratic equations. Specifically, we're going to tackle the equation 7/19x² + 7/19x = 0. Don't worry if that looks a bit intimidating at first glance; we'll break it down step-by-step, making it super easy to understand. We’ll explore the different methods to solve this equation and provide a clear, concise solution. Whether you're a student grappling with algebra or just someone looking to brush up on your math skills, this guide is for you. So, let’s jump right in and make quadratic equations our new best friends!

Understanding Quadratic Equations

Before we dive into solving our specific equation, let's take a moment to understand what quadratic equations are all about. A quadratic equation is a polynomial equation of the second degree. What does that mean? Simply put, it's an equation that includes a term with x squared (x²). The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the equation would become linear, not quadratic.

Now, why are quadratic equations important? Well, they pop up everywhere in the real world! From physics (think projectile motion) to engineering (designing structures) to even economics (modeling growth), quadratic equations help us understand and solve a myriad of problems. So, mastering them is a pretty valuable skill.

Key Components of a Quadratic Equation

Let’s break down the key components of a quadratic equation using our general form, ax² + bx + c = 0:

  • a: This is the coefficient of the x² term. It tells us how much the x² term contributes to the equation. In our example, 7/19x² + 7/19x = 0, the 'a' value is 7/19.
  • b: This is the coefficient of the x term. It tells us how much the x term contributes. In our equation, the 'b' value is also 7/19.
  • c: This is the constant term, a number that stands alone without any x attached. In our equation, we don't see a constant term, which means 'c' is 0. This is a crucial observation that will help us solve the equation.

Understanding these components is the first step in cracking the code of quadratic equations. Now that we have a solid foundation, let's move on to the different methods we can use to solve them.

Methods to Solve Quadratic Equations

There are several methods to solve quadratic equations, each with its strengths and when it's most effective. The most common methods include:

  1. Factoring: This method involves breaking down the quadratic expression into the product of two binomials. It's the quickest method when it works, but it's not always straightforward.
  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial. It's a powerful method that always works, but it can be a bit more involved.
  3. Quadratic Formula: This is the go-to method when factoring or completing the square seems too difficult. The quadratic formula provides a direct solution for x, using the coefficients a, b, and c.

Factoring: A Quick and Efficient Method

Factoring is often the first method we try when solving quadratic equations because it can be the quickest and most efficient. The idea behind factoring is to express the quadratic expression as a product of two simpler expressions (binomials). For example, if we can rewrite ax² + bx + c as (px + q)(rx + s), then we can easily find the solutions for x by setting each factor equal to zero.

  • When to Use Factoring: Factoring is particularly effective when the coefficients 'a', 'b', and 'c' are integers and the quadratic expression can be easily broken down into factors. It's also great when you can quickly spot common factors or patterns.
  • Steps for Factoring:
    1. Look for a Greatest Common Factor (GCF): Always start by checking if there's a common factor that you can factor out from all the terms. This simplifies the equation and makes it easier to work with.
    2. Factor the Quadratic Expression: Once you've factored out the GCF (if there is one), try to factor the remaining quadratic expression into two binomials. This might involve some trial and error, but with practice, you'll get the hang of it.
    3. Set Each Factor to Zero: After factoring, you'll have an equation of the form (px + q)(rx + s) = 0. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero and solve for x.
    4. Solve for x: Solving the resulting equations will give you the solutions for x, also known as the roots of the quadratic equation.

Completing the Square: A Reliable Approach

Completing the square is a powerful technique for solving quadratic equations, and it's particularly useful when the equation is not easily factorable. The main idea behind completing the square is to transform the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root.

  • When to Use Completing the Square: This method is especially handy when the quadratic expression doesn't factor neatly or when you want to derive the quadratic formula. It always works, although it can be a bit more involved than factoring.
  • Steps for Completing the Square:
    1. Ensure 'a' is 1: If the coefficient of x² (the 'a' value) is not 1, divide the entire equation by 'a'.
    2. Move the Constant Term: Move the constant term ('c') to the right side of the equation.
    3. Complete the Square: Take half of the coefficient of the x term (the 'b' value), square it, and add it to both sides of the equation. This step is the heart of the method, as it creates a perfect square trinomial on the left side.
    4. Factor the Trinomial: Factor the perfect square trinomial on the left side into the form (x + k)² or (x - k)², where 'k' is half of the original 'b' value.
    5. Take the Square Root: Take the square root of both sides of the equation, remembering to include both the positive and negative square roots.
    6. Solve for x: Solve the resulting equations for x to find the solutions.

Quadratic Formula: The Universal Solution

The quadratic formula is the ultimate tool in our quadratic equation-solving arsenal. It provides a direct solution for x, regardless of whether the equation can be factored or completed the square easily. The quadratic formula is derived from the method of completing the square, and it's a fundamental formula in algebra.

  • When to Use the Quadratic Formula: This formula is your best friend when factoring is difficult or impossible, and when completing the square seems too cumbersome. It's a universal solution that always works.

  • The Quadratic Formula: The quadratic formula is given by:

    x = (-b ± √(b² - 4ac)) / (2a)

    Where a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0.

  • Steps for Using the Quadratic Formula:

    1. Identify a, b, and c: Determine the values of a, b, and c from the quadratic equation.
    2. Plug into the Formula: Substitute the values of a, b, and c into the quadratic formula.
    3. Simplify: Simplify the expression inside the square root (the discriminant) and then simplify the entire formula.
    4. Solve for x: You'll have two solutions for x, one with the plus sign and one with the minus sign in front of the square root.

Solving 7/19x² + 7/19x = 0: A Step-by-Step Approach

Now that we've covered the main methods for solving quadratic equations, let's apply our knowledge to solve the equation 7/19x² + 7/19x = 0. We'll walk through the solution step-by-step, making sure everything is crystal clear.

Step 1: Identify the Coefficients

First, let's identify the coefficients a, b, and c in our equation. Comparing 7/19x² + 7/19x = 0 to the general form ax² + bx + c = 0, we can see that:

  • a = 7/19
  • b = 7/19
  • c = 0

Notice that the constant term c is 0. This is a key observation that will simplify our solution process.

Step 2: Choose a Solution Method

Given that c = 0, the easiest method to solve this equation is factoring. Factoring is often the most efficient method when the constant term is zero because we can directly factor out a common factor.

Step 3: Factor out the Greatest Common Factor (GCF)

In the equation 7/19x² + 7/19x = 0, the greatest common factor is 7/19x. Let's factor it out:

7/19x(x + 1) = 0

Now we have the equation in a factored form, which makes it easy to find the solutions.

Step 4: Set Each Factor to Zero

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero:

  1. 7/19x = 0
  2. x + 1 = 0

Step 5: Solve for x

Now, let's solve each equation for x:

  1. For 7/19x = 0, we can multiply both sides by 19/7 to isolate x:

    x = 0

    So, our first solution is x₁ = 0.

  2. For x + 1 = 0, we can subtract 1 from both sides to solve for x:

    x = -1

    So, our second solution is x₂ = -1.

Step 6: Verify the Solutions

It's always a good idea to verify our solutions by plugging them back into the original equation:

  1. For x = 0:

    7/19(0)² + 7/19(0) = 0

    0 + 0 = 0

    The equation holds true.

  2. For x = -1:

    7/19(-1)² + 7/19(-1) = 0

    7/19(1) - 7/19 = 0

    7/19 - 7/19 = 0

    The equation holds true.

Both solutions are valid.

Alternative Solutions and Common Mistakes

While factoring was the most straightforward method for solving 7/19x² + 7/19x = 0, let's briefly discuss how we could have used the quadratic formula and also highlight some common mistakes to avoid.

Using the Quadratic Formula

Even though factoring was easier in this case, we can still use the quadratic formula to solve the equation. Remember the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

We identified a = 7/19, b = 7/19, and c = 0. Plugging these values into the formula:

x = (-7/19 ± √((7/19)² - 4(7/19)(0))) / (2(7/19))

Simplifying:

x = (-7/19 ± √(49/361)) / (14/19)

x = (-7/19 ± 7/19) / (14/19)

Now we have two solutions:

  1. x = (-7/19 + 7/19) / (14/19) = 0 / (14/19) = 0
  2. x = (-7/19 - 7/19) / (14/19) = (-14/19) / (14/19) = -1

As you can see, we arrive at the same solutions: x₁ = 0 and x₂ = -1. This confirms that the quadratic formula works, even when factoring is simpler.

Common Mistakes to Avoid

  1. Dividing by x: A common mistake when solving quadratic equations like this is to divide both sides by x. This might seem like a quick way to simplify, but it can lead to the loss of a solution (in this case, x = 0). Always factor instead of dividing by a variable.
  2. Incorrectly Applying the Quadratic Formula: Double-check that you've correctly identified a, b, and c, and that you've substituted them into the formula correctly. Pay close attention to signs and order of operations.
  3. Forgetting the ±: When taking the square root in the quadratic formula or when completing the square, remember to include both the positive and negative square roots. This is crucial for finding both solutions of the quadratic equation.
  4. Not Verifying Solutions: Always verify your solutions by plugging them back into the original equation. This helps catch any errors you might have made during the solving process.

Conclusion: Mastering Quadratic Equations

Great job, guys! We've successfully solved the quadratic equation 7/19x² + 7/19x = 0 using the factoring method. We also saw how the quadratic formula could be used as an alternative, and we discussed common mistakes to avoid. By understanding the different methods and practicing regularly, you can confidently tackle any quadratic equation that comes your way.

Remember, the key to mastering quadratic equations is practice. Try solving different types of quadratic equations using various methods. The more you practice, the more comfortable and confident you'll become. Keep up the great work, and happy solving!