Question

use successive approximations to estimate a solution of this equation: (x - 5)^2 + 3 = 2^{x - 1}. first, complete the table. round your answers to the nearest tenth.

x	(x - 5)^2 + 3	2^{x - 1}
2	12	2
3	7	4
4	4	8
3.4	5.6	5.3
3.5	5.3	5.7
3.45

now, use the table to estimate one solution, rounded to the nearest tenth.
x ≈

Explanation:

Step1: Calculate $(x - 5)^2+3$ for $x = 3.45$

Substitute $x = 3.45$ into $(x - 5)^2+3$. First, calculate $3.45-5=-1.55$. Then $(-1.55)^2 = 2.4025$, and $2.4025 + 3=5.4025\approx5.4$.

Step2: Calculate $2^{x - 1}$ for $x = 3.45$

Substitute $x = 3.45$ into $2^{x - 1}$, we get $2^{3.45-1}=2^{2.45}$. Using a calculator, $2^{2.45}\approx5.5$.

Step3: Estimate the solution

Since the values of $(x - 5)^2+3$ and $2^{x - 1}$ are getting closer as $x$ varies around $3.45$, and looking at the trend in the table, the solution of the equation $(x - 5)^2+3 = 2^{x - 1}$ rounded to the nearest tenth is $x\approx3.5$.

Answer:

For $x = 3.45$, $(x - 5)^2+3\approx5.4$, $2^{x - 1}\approx5.5$, and the estimated solution of the equation is $x\approx3.5$