Finding X1^3 + X2^3 Given Roots Of X^2 + X - 1 = 0

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Hey guys! Let's dive into a fun algebra problem today. We're going to figure out how to find the value of x1^3 + x2^3 when x1 and x2 are the roots of the quadratic equation x^2 + x - 1 = 0. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step and make it super clear. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the question is asking. We are given a quadratic equation, which is x^2 + x - 1 = 0. Remember, a quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The roots of a quadratic equation are the values of x that make the equation true. In this case, we're told that x1 and x2 are these roots. Our mission is to find the value of x1^3 + x2^3. This means we need to find the sum of the cubes of these roots. Sounds like a plan, right? To effectively solve this, we'll need to leverage our knowledge of quadratic equations, roots, and some handy algebraic identities. Keep in mind the relationship between the roots and the coefficients of the quadratic equation, which will be our key to unlocking the solution. So, stay with me, and let's see how we can crack this problem!

Key Concepts: Roots and Coefficients

Okay, so how do we connect the roots of the equation to the expression we want to find? This is where the relationship between roots and coefficients of a quadratic equation comes into play. For any quadratic equation ax^2 + bx + c = 0, there are some cool formulas that link the roots (x1 and x2) to the coefficients (a, b, and c). Specifically:

  1. The sum of the roots: x1 + x2 = -b/a
  2. The product of the roots: x1 * x2 = c/a

These two formulas are super important for solving problems like this. They give us a way to talk about the roots without actually having to calculate them individually. Now, let's apply these formulas to our specific equation, x^2 + x - 1 = 0. In this case, a = 1, b = 1, and c = -1. Using the formulas, we can quickly find:

  • x1 + x2 = -1/1 = -1
  • x1 * x2 = -1/1 = -1

So, we know the sum and product of the roots. Great! But how does this help us find x1^3 + x2^3? That's the next piece of the puzzle, and it involves another useful algebraic identity. Let's move on and see how we can use this information to get closer to our answer!

The Magic Identity: Sum of Cubes

Alright, we've got the sum and product of the roots, but we need the sum of their cubes. This is where our friendly neighborhood algebraic identity for the sum of cubes comes to the rescue! The identity is:

a^3 + b^3 = (a + b)^3 - 3ab(a + b)

This might look a bit intimidating, but trust me, it's a powerful tool. Notice how it relates the sum of cubes (a^3 + b^3) to the sum (a + b) and product (ab) of the numbers. This is exactly what we need! We can apply this identity to our problem by simply substituting x1 for a and x2 for b. This gives us:

x1^3 + x2^3 = (x1 + x2)^3 - 3x1x2(x1 + x2)

Now, look at that! We have an expression for x1^3 + x2^3 that only involves the sum (x1 + x2) and product (x1 * x2) of the roots. And guess what? We already know those values! We found earlier that x1 + x2 = -1 and x1 * x2 = -1. So, we're ready to plug these values into the identity and finally calculate the value of x1^3 + x2^3. Let's do it!

Calculating x1^3 + x2^3

Okay, it's time to put everything together and actually calculate x1^3 + x2^3. We have the identity:

x1^3 + x2^3 = (x1 + x2)^3 - 3x1x2(x1 + x2)

And we know that:

  • x1 + x2 = -1
  • x1 * x2 = -1

So, let's substitute these values into the identity:

x1^3 + x2^3 = (-1)^3 - 3(-1)(-1)

Now, let's simplify this step by step. First, (-1)^3 is just -1. Then, 3(-1)(-1) is 3 * 1, which is 3. So, our equation becomes:

x1^3 + x2^3 = -1 - 3

And finally, -1 - 3 = -4.

So, we've found it! The value of x1^3 + x2^3 is -4. That's pretty cool, right? We started with a quadratic equation and, by using some clever tricks and identities, we were able to find the sum of the cubes of its roots. You might be asking what exact root values are, but with the method, we do not need to find each root value independently.

Recap and Final Thoughts

Wow, we've covered a lot! Let's quickly recap what we did. We started with the problem of finding x1^3 + x2^3 where x1 and x2 are the roots of x^2 + x - 1 = 0. We used the relationship between the roots and coefficients of a quadratic equation to find that x1 + x2 = -1 and x1 * x2 = -1. Then, we pulled out the sum of cubes identity:

a^3 + b^3 = (a + b)^3 - 3ab(a + b)

We substituted our values into this identity and, with a bit of arithmetic, found that x1^3 + x2^3 = -4. So, the answer is E) -4. Isn't it amazing how we can solve seemingly complex problems by breaking them down into smaller, manageable steps and using the right tools? Remember, algebra is like a toolbox full of cool tricks and identities. The more you practice, the better you'll become at choosing the right tool for the job. You can also verify if each root is equal to x1 = (-1 + √5) / 2 and x2 = (-1 - √5) / 2, then calculate the value of x1^3 + x2^3. Great job, guys! Keep up the awesome work, and I'll see you next time with another fun math problem!