To find the values of [tex]\( y \)[/tex] corresponding to the given [tex]\( x \)[/tex] values based on the equation [tex]\( y = x^2 - 10 \)[/tex]:
### Step-by-Step Solution:
1. For [tex]\( x = -6 \)[/tex]:
Substitute [tex]\( x = -6 \)[/tex] into the equation:
[tex]\[
y = (-6)^2 - 10
\][/tex]
Calculating the value:
[tex]\[
y = 36 - 10 = 26
\][/tex]
So, [tex]\( y = 26 \)[/tex] when [tex]\( x = -6 \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
Substitute [tex]\( x = -1 \)[/tex] into the equation:
[tex]\[
y = (-1)^2 - 10
\][/tex]
Calculating the value:
[tex]\[
y = 1 - 10 = -9
\][/tex]
So, [tex]\( y = -9 \)[/tex] when [tex]\( x = -1 \)[/tex].
3. For [tex]\( x = 0 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = 0^2 - 10
\][/tex]
Calculating the value:
[tex]\[
y = 0 - 10 = -10
\][/tex]
So, [tex]\( y = -10 \)[/tex] when [tex]\( x = 0 \)[/tex].
4. Given [tex]\( y = 6 \)[/tex]:
Substitute the known [tex]\( y \)[/tex] value [tex]\( y = 6 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
6 = x^2 - 10
\][/tex]
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 = 6 + 10
\][/tex]
[tex]\[
x^2 = 16
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm 4
\][/tex]
Since [tex]\( x \)[/tex] was not specified, one possible value for [tex]\( x \)[/tex] could be [tex]\( 4 \)[/tex] or [tex]\(-4\)[/tex].
5. Given [tex]\( y = 90 \)[/tex]:
Substitute the known [tex]\( y \)[/tex] value [tex]\( y = 90 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
90 = x^2 - 10
\][/tex]
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 = 90 + 10
\][/tex]
[tex]\[
x^2 = 100
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm 10
\][/tex]
Since [tex]\( x \)[/tex] was not specified, one possible value for [tex]\( x \)[/tex] could be [tex]\( 10 \)[/tex] or [tex]\(-10\)[/tex].
Combining all these, the completed table with the missing [tex]\( y \)[/tex] values is:
[tex]\[
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$x$ & -6 & -1 & 0 & 4? & 10? \\
\hline
$y$ & 26 & -9 & -10 & 6 & 90 \\
\hline
\end{tabular}
\][/tex]
The calculated missing [tex]\( y \)[/tex] values for the given [tex]\( x \)[/tex] values are:
### Detailed [tex]\(\ y\)[/tex] Values:
- When [tex]\( x = -6 \)[/tex], [tex]\( y = 26 \)[/tex].
- When [tex]\( x = -1 \)[/tex], [tex]\( y = -9 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -10 \)[/tex].
These values match the obtained results when solving the problem step by step.
