Solving Equations: A Comprehensive Guide With Examples

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Hey everyone! Let's dive into the world of solving equations. It might seem a little daunting at first, but trust me, with a bit of practice and the right approach, you'll be cracking these problems like a pro. In this guide, we'll tackle some specific examples, breaking down each step so you can follow along easily. We'll cover different types of equations and the methods to solve them, so you'll be well-equipped to handle various problems. Solving equations is a fundamental skill in algebra, so getting a good grasp of it will set you up for success in your math journey! Let's get started.

Equation №2: 2x³ - x² + 6x - 3 = 0

Alright, let's tackle our first equation: 2x³ - x² + 6x - 3 = 0. This one looks a bit more complex because it's a cubic equation (highest power of x is 3). The key here is to recognize patterns and use techniques to simplify it. Notice that we have four terms, which often hints at factoring by grouping. So, let's give that a shot!

Factoring by Grouping

  1. Group the terms: We'll group the first two terms and the last two terms together: (2x³ - x²) + (6x - 3) = 0. This is like putting them into separate little bundles.
  2. Factor out the greatest common factor (GCF) from each group:
    • For the first group (2x³ - x²), the GCF is x². Factoring it out, we get x²(2x - 1).
    • For the second group (6x - 3), the GCF is 3. Factoring it out, we get 3(2x - 1). So, now our equation looks like this: x²(2x - 1) + 3(2x - 1) = 0.
  3. Notice the common factor: See how both terms now have a common factor of (2x - 1)? That's the magic of grouping!
  4. Factor out the common binomial: We can factor out (2x - 1) from both terms: (2x - 1)(x² + 3) = 0. This is a huge step because we've reduced a complex equation into a product of simpler factors.

Solving for x

Now that we've factored the equation, we can solve for x by setting each factor equal to zero:

  1. Set the first factor to zero: 2x - 1 = 0.
    • Add 1 to both sides: 2x = 1.
    • Divide both sides by 2: x = 1/2.
  2. Set the second factor to zero: x² + 3 = 0.
    • Subtract 3 from both sides: x² = -3.
    • Take the square root of both sides: x = ±√(-3). Since the square root of a negative number is imaginary, we get x = ±i√3, where i is the imaginary unit (i² = -1).

Solutions:

So, the solutions to the equation 2x³ - x² + 6x - 3 = 0 are: x = 1/2, x = i√3, and x = -i√3. We have one real solution and two complex solutions. Congrats, you've solved a cubic equation!

Keep in mind that when solving equations, especially those with higher degrees, there might be real and complex solutions. The number of solutions often matches the highest power of the variable (in this case, x).

Equation №3: 5x³ + 20x = 0

Let's move on to our next equation: 5x³ + 20x = 0. This one's a bit different, but don't worry, it's still manageable! The key is to recognize the common factor and simplify from there. Let's dive in!

Factoring Out the GCF

  1. Identify the Greatest Common Factor: Both terms, 5x³ and 20x, have a common factor of 5x. This is the biggest term that divides evenly into both terms.
  2. Factor out the GCF: We factor out 5x from both terms:
    • 5x³ becomes 5x * x²
    • 20x becomes 5x * 4
    • So the equation 5x³ + 20x = 0 transforms into 5x(x² + 4) = 0. Much simpler, right?

Solving for x

Now that we have the factored form, we can solve for x by setting each factor equal to zero:

  1. Set the first factor to zero: 5x = 0.
    • Divide both sides by 5: x = 0.
  2. Set the second factor to zero: x² + 4 = 0.
    • Subtract 4 from both sides: x² = -4.
    • Take the square root of both sides: x = ±√(-4). Since we have a negative number under the square root, we'll get imaginary solutions. x = ±2i, where i is the imaginary unit (i² = -1).

Solutions:

The solutions to the equation 5x³ + 20x = 0 are: x = 0, x = 2i, and x = -2i. This equation has one real solution (x = 0) and two imaginary solutions. Recognizing common factors is critical when solving equations, allowing you to significantly reduce the complexity of the problem and simplify the process.

Equation №4: 5x³ + 20x = 0

Surprise, we have the same equation as before: 5x³ + 20x = 0. This is great practice! Let's run through the steps again to solidify our understanding. This repeated example provides an excellent opportunity to reinforce the solving equations process and cement our knowledge.

Factoring Out the GCF (Again!)

  1. Identify the Greatest Common Factor: Just like before, both terms, 5x³ and 20x, share a greatest common factor of 5x.
  2. Factor out the GCF: We factor 5x out of both terms:
    • 5x³ becomes 5x * x²
    • 20x becomes 5x * 4
    • The equation 5x³ + 20x = 0 becomes 5x(x² + 4) = 0.

Solving for x (Again!)

Now, let's solve for x by setting each factor equal to zero:

  1. Set the first factor to zero: 5x = 0.
    • Divide both sides by 5: x = 0.
  2. Set the second factor to zero: x² + 4 = 0.
    • Subtract 4 from both sides: x² = -4.
    • Take the square root of both sides: x = ±√(-4). This gives us imaginary solutions: x = ±2i.

Solutions (Again!)

The solutions to the equation 5x³ + 20x = 0 are: x = 0, x = 2i, and x = -2i. Again, one real solution (x = 0) and two imaginary solutions. This repetition helps reinforce the steps involved and highlights how consistently factoring and solving can be used. Each time we practice solving equations, the steps become more natural and the process becomes faster. Repetition is a key to mastering these concepts! We successfully factored out the GCF, solved for x, and identified both real and complex solutions.