To find the solutions to the equation [tex]\(10^{2x} + 11 = (x+6)^2 - 2\)[/tex], let's break down the problem step by step and analyze for which [tex]\( x \)[/tex] values the equation holds.
1. Understand the equation:
[tex]\[
10^{2x} + 11 = (x+6)^2 - 2
\][/tex]
2. Rewrite the equation for better clarity:
[tex]\[
10^{2x} + 11 = (x+6)^2 - 2
\][/tex]
[tex]\[
10^{2x} + 11 = x^2 + 12x + 36 - 2
\][/tex]
[tex]\[
10^{2x} + 11 = x^2 + 12x + 34
\][/tex]
3. Isolate the terms involving [tex]\( x \)[/tex]:
[tex]\[
10^{2x} - x^2 - 12x - 23 = 0
\][/tex]
The task now is to determine the values of [tex]\( x \)[/tex] that satisfy this equation. After some consideration, we conclude with some approximate values through numerical methods.
4. List and check the approximate solutions:
Comparing the values obtained with the equation:
- [tex]\( x = -9.6 \)[/tex]
[tex]\[
10^{2(-9.6)} + 11 \approx ( -9.6 + 6 )^2 - 2
\][/tex]
[tex]\[
10^{-19.2} + 11 \approx ( -3.6 )^2 - 2
\][/tex]
[tex]\[
0 \approx 12.96 - 2
\][/tex]
[tex]\[
0 \approx 10.96
\][/tex]
Hence, [tex]\( x = -9.6 \approx -9.605551275463988 \)[/tex]
- [tex]\( x = -7.4 \)[/tex]
[tex]\[
10^{2(-7.4)} + 11 = ( -7.4 + 6 )^2 - 2
\][/tex]
[tex]\[
10^{-14.8} + 11 \approx (-1.4)^2 - 2
\][/tex]
[tex]\[
0 + 11 \approx 1.96 - 2
\][/tex]
[tex]\[
11 \approx -0.04 \quad \text{(Not a solution)}
\][/tex]
- [tex]\( x = -4.6 \)[/tex]
[tex]\[
10^{2(-4.6)} + 11 = ( -4.6 + 6 )^2 - 2
\][/tex]
[tex]\[
10^{-9.2} + 11 \approx ( 1.4 )^2 - 2
\][/tex]
[tex]\[
0 + 11 \approx 1.96 - 2
\][/tex]
[tex]\[
11 \approx -0.04 \quad \text{(Not a solution)}
\][/tex]
- [tex]\( x = -2.4 \)[/tex]
[tex]\[
10^{2(-2.4)} + 11 = ( -2.4 + 6 )^2 - 2
\][/tex]
[tex]\[
10^{-4.8} + 11 \approx ( 3.6 )^2 - 2
\][/tex]
[tex]\[
0 + 11 \approx 12.96 - 2
\][/tex]
[tex]\[
11 \approx 10.96
\][/tex]
Hence, [tex]\( x = -2.4 \approx -2.394446469750409 \)[/tex]
- [tex]\( x = 0.6 \)[/tex]
[tex]\[
10^{2(0.6)} + 11 = ( 0.6 + 6 )^2 - 2
\][/tex]
[tex]\[
10^{1.2} + 11 = ( 6.6 )^2 - 2
\][/tex]
[tex]\[
\approx 15.85 + 11 = 43.56 - 2
\][/tex]
[tex]\[
26.85 \approx 41.56 \quad \text{(Not a solution)}
\][/tex]
We conclude that the approximate solutions to the equation [tex]\(10^{2x} + 11 = (x+6)^2 - 2\)[/tex] are [tex]\(x = -9.6\)[/tex] and [tex]\( x = -2.4\)[/tex].
So, the two appropriate values are:
- [tex]\(-9.6\)[/tex]
- [tex]\(-2.4\)[/tex]
