Question

use successive approximations to estimate a solution of this equation:
\\(0.5^x = 2.5x^2 - 1\\)
first, complete the table. round your answers to the nearest tenth.

\\(x\\)	\\(0.5^x\\)	\\(2.5x^2 - 1\\)
\\(-2\\)	\\(4\\)	\\(9\\)
\\(-1\\)	\\(2\\)	\\(1.5\\)
\\(-1.2\\)	\\(2.3\\)	\\(2.6\\)
\\(-1.1\\)	\\(2.1\\)	\\(2\\)
\\(-1.15\\)	\\(3\\)	\\(\square\\)

now, use the table to estimate one solution, rounded to the nearest tenth.
\\(x \approx \square\\)

Explanation:

Step1: Calculate $0.5^{-1.15}$

$0.5^{-1.15} = (2^{-1})^{-1.15} = 2^{1.15} \approx 2.2$

Step2: Calculate $2.5(-1.15)^2 -1$

First compute $(-1.15)^2 = 1.3225$, then $2.5 \times 1.3225 = 3.30625$, finally $3.30625 - 1 = 2.3$

Step3: Compare values to estimate solution

At $x=-1.1$, $0.5^x=2.1$ and $2.5x^2-1=2$; at $x=-1.2$, $0.5^x=2.3$ and $2.5x^2-1=2.6$. The values cross between $x=-1.1$ and $x=-1.2$. At $x=-1.15$, $0.5^x \approx 2.2$ and $2.5x^2-1 \approx 2.3$, which are closest, so round to nearest tenth.

Answer:

Completed table values:
$0.5^{-1.15} \approx 2.2$, $2.5(-1.15)^2 -1 \approx 2.3$
Estimated solution: $x \approx -1.2$