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Session
x
4. Cora is using successive approximations to estimate a positive solution to
f(x)=g(x), where
f(x)=x^(2)+13 and
g(x)=3x+14. The table shows her results for different input values of
x.

x

f(x)

g(x)

0
13
14

1
14
17

2
17
20

3
22
23

4
29
26

3.5
25.25
24.5

Use Cora's process to find the positive solution, to the nearest tenth, of
f(x)=g(x).

Session $x$ $\newline$ $4$ . Cora is using successive approximations to estimate a positive solution to $f(x)=g(x)$ , where $f(x)=x^{2}+13$ and $g(x)=3 x+14$ . The table shows her results for different input values of $x$ . $\newline$ \begin{tabular}{|c|c|c|} $\newline$ \hline $x$ & $f(x)$ & $g(x)$ \\ $\newline$ \hline $0$ & $13$ & $14$ \\ $\newline$ \hline $1$ & $14$ & $17$ \\ $\newline$ \hline $2$ & $17$ & $20$ \\ $\newline$ \hline $3$ & $22$ & $23$ \\ $\newline$ \hline $4$ & $29$ & $26$ \\ $\newline$ \hline $3$ . $5$ & $25$ . $25$ & $24$ . $5$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ Use Cora's process to find the positive solution, to the nearest tenth, of $f(x)=g(x)$ .

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Writing from expanded to exponent form

Full solution

Q. Session

x

\newline

4

. Cora is using successive approximations to estimate a positive solution to

f(x)=g(x)

, where

f(x)=x^{2}+13

and

g(x)=3 x+14

. The table shows her results for different input values of

x

.

\newline

\begin{tabular}{|c|c|c|}

\newline

\hline

x

&

f(x)

&

g(x)

\\

\newline

\hline

0

&

13

&

14

\\

\newline

\hline

1

&

14

&

17

\\

\newline

\hline

2

&

17

&

20

\\

\newline

\hline

3

&

22

&

23

\\

\newline

\hline

4

&

29

&

26

\\

\newline

\hline

3

.

5

&

25

.

25

&

24

.

5

\\

\newline

\hline

\newline

\end{tabular}

\newline

Use Cora's process to find the positive solution, to the nearest tenth, of

f(x)=g(x)

.

Analyze the table: Step $1$ : Analyze the table to find where $f(x)$ and $g(x)$ values are closest. $\newline$ - $f(3) = 22$ , $g(3) = 23$ $\newline$ - $f(3.5) = 25.25$ , $g(3.5) = 24.5$ $\newline$ - $f(4) = 29$ , $g(4) = 26$ $\newline$ The values are closest between $x = 3$ and $x = 4$ .
Interpolation estimate: Step $2$ : Use interpolation to estimate the solution between $x = 3$ and $x = 4$ .
- Slope of $f(x)$ from $x = 3$ to $x = 4$ : $(29 - 22) / (4 - 3) = 7$
- Slope of $g(x)$ from $x = 3$ to $x = 4$ : $(26 - 23) / (4 - 3) = 3$
- Difference in slopes = $x = 4$ $0$
- Difference in function values at $x = 3$ : $x = 4$ $2$
- Estimate $x = 4$ $3$ where $x = 4$ $4$ : $x = 4$ $5$ , so $x = 4$ $6$

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