Solving The Quadratic Equation: X^2 = 4x - 14

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Hey guys! Let's dive into solving this quadratic equation: x^2 = 4x - 14. Quadratic equations might seem intimidating at first, but don't worry, we'll break it down step by step. We'll explore different methods to tackle this problem, ensuring you understand the process thoroughly. By the end of this guide, you'll be equipped to solve similar equations with confidence. So, let's put on our math hats and get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let’s quickly recap what quadratic equations are all about. A quadratic equation is a polynomial equation of the second degree. This basically means that the highest power of the variable (in our case, 'x') is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it would be a linear equation). Our goal when solving a quadratic equation is to find the values of 'x' that satisfy the equation. These values are also known as the roots or solutions of the equation.

Now, looking at our equation, x^2 = 4x - 14, we need to rearrange it into the standard form to make it easier to work with. Subtracting 4x and adding 14 to both sides gives us:

x² - 4x + 14 = 0

Now that we have it in the standard form, we can clearly see that a = 1, b = -4, and c = 14. With this sorted, let’s explore the methods we can use to solve it. There are generally three main methods for solving quadratic equations:

  1. Factoring
  2. Completing the square
  3. Using the quadratic formula

Let's go through each of these and see which one works best for our equation.

Method 1: Factoring

Factoring involves expressing the quadratic expression as a product of two linear factors. This method is efficient when the quadratic equation can be factored easily. However, not all quadratic equations can be factored using simple integers, which might be the case with our equation. But let’s try it out anyway!

To factor the quadratic equation x² - 4x + 14 = 0, we need to find two numbers that multiply to 14 (the constant term) and add up to -4 (the coefficient of the x term). Let's think about the factor pairs of 14:

  • 1 and 14
  • 2 and 7

None of these pairs, or their negative counterparts, add up to -4. This indicates that our quadratic equation cannot be easily factored using integers. So, while factoring is a great method when it works, it's not the right tool for this particular job. Don't worry, we have other methods in our toolbox!

Method 2: Completing the Square

Completing the square is a powerful technique that can be used to solve any quadratic equation. It involves transforming the equation into a perfect square trinomial, which can then be easily solved by taking the square root. This method might seem a bit tricky at first, but with practice, you’ll get the hang of it.

Here's how we can apply completing the square to our equation, x² - 4x + 14 = 0:

  1. Move the constant term to the right side of the equation: x² - 4x = -14

  2. Take half of the coefficient of the x term (which is -4), square it, and add it to both sides of the equation. Half of -4 is -2, and (-2)² is 4. So, we add 4 to both sides: x² - 4x + 4 = -14 + 4 x² - 4x + 4 = -10

  3. Rewrite the left side as a squared binomial. The left side is now a perfect square trinomial, which can be factored as (x - 2)²: (x - 2)² = -10

  4. Take the square root of both sides: √(x - 2)² = ±√(-10) x - 2 = ±√(-10)

  5. Solve for x: x = 2 ± √(-10)

Notice that we have the square root of a negative number, which means our solutions will be complex numbers. We can rewrite √(-10) as √(10 * -1) = √(10) * √(-1) = √(10)i, where 'i' is the imaginary unit (√(-1)).

So, our solutions are:

x = 2 ± √(10)i

Thus, completing the square gave us complex solutions, which is perfectly fine! It just means the parabola represented by the quadratic equation doesn’t intersect the x-axis. Completing the square can be a bit lengthy, but it's a solid method, especially when factoring doesn't work. Now, let’s look at the quadratic formula, which is like the ultimate Swiss Army knife for solving quadratic equations.

Method 3: Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. It's derived from the process of completing the square, but it provides a direct way to find the solutions without going through all the steps. The formula is:

x = [-b ± √(b² - 4ac)] / (2a)

Remember our equation, x² - 4x + 14 = 0? We identified a = 1, b = -4, and c = 14 earlier. Now, let's plug these values into the quadratic formula:

x = [-(-4) ± √((-4)² - 4 * 1 * 14)] / (2 * 1)

Let's simplify this step by step:

x = [4 ± √(16 - 56)] / 2 x = [4 ± √(-40)] / 2

Again, we encounter the square root of a negative number, indicating complex solutions. We can simplify √(-40) as √(40 * -1) = √(40) * √(-1) = √(4 * 10) * i = 2√(10)i.

So, our equation becomes:

x = [4 ± 2√(10)i] / 2

Now, we can divide both terms in the numerator by 2:

x = 2 ± √(10)i

Look at that! We arrived at the same solutions as when we completed the square. This confirms that our solutions are correct. The quadratic formula is super reliable and always works, even when dealing with complex roots. It's a fantastic tool to have in your math arsenal.

Comparing the Methods

So, we’ve solved x² = 4x - 14 using completing the square and the quadratic formula. We saw that factoring didn't work in this case, which highlights an important point: not all quadratic equations can be easily factored. But when factoring is possible, it's often the quickest method.

Completing the square is a robust method and helps in understanding the derivation of the quadratic formula. However, it can be a bit more involved and requires careful manipulation of the equation.

The quadratic formula is the most versatile method. It works for any quadratic equation, regardless of whether the solutions are real or complex. It's a straightforward plug-and-chug approach, making it a reliable choice for most situations.

In this specific case, both completing the square and the quadratic formula gave us the same complex solutions: x = 2 ± √(10)i. This reinforces the idea that different methods can lead to the same correct answer, which is always a satisfying outcome.

Conclusion

We’ve successfully solved the quadratic equation x² = 4x - 14 using multiple methods! We explored factoring, completing the square, and the quadratic formula. Factoring turned out to be unsuitable for this equation, but both completing the square and the quadratic formula led us to the complex solutions: x = 2 ± √(10)i.

Remember, guys, the key to mastering quadratic equations is practice. Try solving different equations using these methods to build your confidence and skills. Each method has its strengths, and knowing when to use which one is a crucial part of problem-solving. So, keep practicing, and you'll become a quadratic equation-solving pro in no time! Happy math-ing!