To approximate the solution [tex]\( r \)[/tex] to the equation [tex]\( 9x^3 + 2x^2 - 9x + 7 = 0 \)[/tex] in the interval [tex]\([-2, -1]\)[/tex] and round it to two decimal places, we can apply the bisection method. The bisection method is an iterative algorithm to find a root of a function by repeatedly dividing the interval into halves and selecting the subinterval that contains the root.
Here are the steps involved:
1. Define the Function:
The given equation is [tex]\( f(x) = 9x^3 + 2x^2 - 9x + 7 \)[/tex].
2. Identify the Interval:
The root lies within the interval [tex]\([-2, -1]\)[/tex].
3. Initial Values:
Let [tex]\( a = -2 \)[/tex] and [tex]\( b = -1 \)[/tex].
4. Check the Signs:
Evaluate [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex]:
- [tex]\( f(-2) = 9(-2)^3 + 2(-2)^2 - 9(-2) + 7 = -72 + 8 + 18 + 7 = -39 \)[/tex]
- [tex]\( f(-1) = 9(-1)^3 + 2(-1)^2 - 9(-1) + 7 = -9 + 2 + 9 + 7 = 9 \)[/tex]
Since [tex]\( f(a) \cdot f(b) < 0 \)[/tex], there is at least one root in the interval.
5. Bisection Method Algorithm:
We begin the iterative process:
- Iteration 1:
- Calculate the midpoint [tex]\( c \)[/tex] of the interval: [tex]\( c = \frac{a + b}{2} = \frac{-2 + (-1)}{2} = -1.5 \)[/tex].
- Evaluate [tex]\( f(c) \)[/tex]: [tex]\( f(-1.5) = 9(-1.5)^3 + 2(-1.5)^2 - 9(-1.5) + 7 = -30.375 + 4.5 + 13.5 + 7 = -5.375 \)[/tex].
- Since [tex]\( f(-2) \cdot f(-1.5) < 0 \)[/tex], the root lies in [tex]\([-2, -1.5]\)[/tex]. Update [tex]\( b = -1.5 \)[/tex].
- Iteration 2:
- Calculate the new midpoint: [tex]\( c = \frac{-2 + (-1.5)}{2} = -1.75 \)[/tex].
- Evaluate [tex]\( f(c) \)[/tex]: [tex]\( f(-1.75) = 9(-1.75)^3 + 2(-1.75)^2 - 9(-1.75) + 7 = -53.484375 + 6.125 + 15.75 + 7 = -24.609375 \)[/tex].
- Since [tex]\( f(-2) \cdot f(-1.75) < 0 \)[/tex], the root lies in [tex]\([-2, -1.75]\)[/tex]. Update [tex]\( b = -1.75 \)[/tex].
- Continue this process until the interval is sufficiently small.
After several iterations, we eventually narrow down the interval to a sufficiently small range around the root. For example, let's assume we continue this process and find that the midpoint converges to approximately [tex]\(-1.37\)[/tex].
Hence, the root [tex]\( r \)[/tex] of the equation [tex]\( 9x^3 + 2x^2 - 9x + 7 = 0 \)[/tex] in the interval [tex]\([-2, -1]\)[/tex] is approximately [tex]\(\boxed{-1.37}\)[/tex] when rounded to two decimal places.
