To solve the equation [tex]\(x^2 + 9x - 13 = 0\)[/tex] graphically using the graph of [tex]\(y = x^2 + 7x - 5\)[/tex], follow these steps:
1. Rewrite the equation to find the difference:
Begin by subtracting the given parabola equation [tex]\(y = x^2 + 7x - 5\)[/tex] from the quadratic equation [tex]\(x^2 + 9x - 13 = 0\)[/tex]. This subtraction isolates the first equation and reveals the line needed.
Here's the subtraction:
[tex]\[
(x^2 + 9x - 13) - (x^2 + 7x - 5)
\][/tex]
2. Simplify the difference:
Simplify the expression from the subtraction, step by step:
[tex]\[
x^2 + 9x - 13 - x^2 - 7x + 5
\][/tex]
Combine like terms:
[tex]\[
(x^2 - x^2) + (9x - 7x) + (-13 + 5)
\][/tex]
This simplification yields:
[tex]\[
0x + 2x - 8
\][/tex]
Thus, we get:
[tex]\[
2x - 8
\][/tex]
3. Express the result as a line equation:
The result is already in the form of a linear equation with the format [tex]\(y = mx + c\)[/tex].
In our case:
[tex]\[
y = 2x - 8
\][/tex]
4. Conclusion:
Therefore, the line you should draw on the graph of [tex]\(y = x^2 + 7x - 5\)[/tex] to find the solutions of [tex]\(x^2 + 9x - 13 = 0\)[/tex] graphically is:
[tex]\[
y = 2x - 8
\][/tex]
Here's the final answer, which is the equation of the line:
[tex]\[
y = 2x - 8
\][/tex]
