Question

use successive approximations to estimate a solution of this equation: 4(0.5)^x=-3x^2 + 14. first, complete the table. round your answers to the nearest tenth. now, use the table to estimate one solution, rounded to the nearest tenth.

Explanation:

Step1: Calculate $4(0.5)^{-1.25}$

$4\times(0.5)^{-1.25}=4\times2^{1.25}\approx4\times2.3784\approx9.5$

Step2: Calculate $-3x^{2}+14$ for $x = - 1.25$

$-3\times(-1.25)^{2}+14=-3\times1.5625 + 14=-4.6875+14\approx9.3$

Step3: Estimate the solution

Looking at the table values, when $x=-1.2$, the values of $4(0.5)^{x}$ and $-3x^{2}+14$ are very close. So the estimated solution rounded to the nearest tenth is $x\approx - 1.2$.

Answer:

For $x=-1.25$, the first - blank is approximately $9.5$ and the second - blank is approximately $9.3$. The estimated solution is $x\approx - 1.2$