Th equation x^3 - x^2 - 6 = 0 has one real root, denoted by a.
Find by calculation the pair of consecutive integers between a lies.
f(2) = 8-4-6 = -2
f(3) = 27-9-6 = 12
So, a root is between 2 and 3.
Well, let's see if we can "calculate" our way to finding the pair of consecutive integers between the real root a.
Using the equation given: x^3 - x^2 - 6 = 0, we can try some good old-fashioned guesswork to find the value of a.
By plugging in a few values for x, we find that when x = 2, the equation becomes 2^3 - 2^2 - 6 = 8 - 4 - 6 = -2. And when x = 3, the equation becomes 3^3 - 3^2 - 6 = 27 - 9 - 6 = 12.
Since we're trying to find the value of a that makes the equation equal to zero, it looks like a is somewhere in between 2 and 3.
Therefore, the pair of consecutive integers between which a lies is 2 and 3. Voila!
But hey, if you want a more "calculating" answer, you can always use numerical methods like bisection or Newton's method to find a more precise value for a. Just be careful not to fall into a numerical rabbit hole!
To find the pair of consecutive integers between which the real root, denoted by a, lies, we can use the method of trial and error.
First, let's substitute some values to determine the sign of the equation for different integers.
When we substitute x = 1, we get:
1^3 - 1^2 - 6 = 1 - 1 - 6 = -6
Since the result is negative, it means that the real root lies to the right of 1.
Next, let's substitute x = 2:
2^3 - 2^2 - 6 = 8 - 4 - 6 = -2
The result is still negative. This tells us that the real root is between 1 and 2.
Now, let's substitute x = 3:
3^3 - 3^2 - 6 = 27 - 9 - 6 = 12
The result is positive. This suggests that the real root lies between 2 and 3.
Continuing this process, let's substitute x = 2.5:
2.5^3 - 2.5^2 - 6 = 15.625 - 6.25 - 6 = 3.375
The result is positive. This implies that the real root lies between 2.5 and 3.
Finally, let's substitute x = 2.75:
2.75^3 - 2.75^2 - 6 = 25.796875 - 7.5625 - 6 = 12.234375
The result is still positive. This indicates that the real root lies between 2.5 and 2.75.
Therefore, based on our calculations, the pair of consecutive integers between which the real root, denoted by a, lies are 2 and 3.
To find the pair of consecutive integers between which the real root of the equation x^3 - x^2 - 6 = 0 lies, you can use the following steps:
Step 1: Solve the equation x^3 - x^2 - 6 = 0 to find the value of the real root, denoted by a.
In this case, the equation is a cubic equation, which can be solved using various methods such as factoring, synthetic division, or numerical methods. Let's use the factoring method for simplicity.
To factor the equation, we can express it as the product of linear factors. However, it is not easy to find the roots by factoring directly, so we can try to use synthetic division to find a factor or factor theorem to find a root.
Using synthetic division or factoring, we can find that one of the roots of the equation is x = -2.
Step 2: Set up the inequality using the root found in Step 1.
Since the root is x = -2, we can set up the inequality -2 < a < -1 to find the pair of consecutive integers between which the real root lies.
Step 3: Determine the pair of consecutive integers.
From the inequality -2 < a < -1, we can conclude that the pair of consecutive integers between which the real root of the equation x^3 - x^2 - 6 = 0 lies are -2 and -1.
Therefore, the pair of consecutive integers is (-2, -1).