To solve the equation $-x-3=x^2-2 x-15$, you could graph the following system:

[tex]\begin{array}{l}
y=-x-3 \
y=x^2-2 x-15\end{array}[/tex]

Use the graph of this system to identify all solutions of $-x-3=x^2-2 x-15$.

A. -7
B. -3
C. 0
D. 4

We have a system of equations y = − x − 3 and y = x 2 − 2 x − 15 .
We test the given x-values to see if they satisfy both equations.
x = − 3 gives y = 0 for both equations, so it's a solution.
x = 4 gives y = − 7 for both equations, so it's a solution. The solutions are − 3 , 4 ​ .

Explanation

Understanding the Problem We are given the system of equations:

{ y = − x − 3 y = x 2 − 2 x − 15 ​
We need to find the solutions to the equation − x − 3 = x 2 − 2 x − 15 by identifying the x-coordinates of the intersection points of the two graphs. The possible solutions are -7, -3, 0, and 4.

Solution Strategy We will substitute each of the given x-values (-7, -3, 0, and 4) into both equations of the system and check if the y-values obtained from both equations are equal.

Checking Possible Solutions

For x = − 7 :

y = − ( − 7 ) − 3 = 7 − 3 = 4
y = ( − 7 ) 2 − 2 ( − 7 ) − 15 = 49 + 14 − 15 = 48 Since 4  = 48 , x = − 7 is not a solution.


For x = − 3 :

y = − ( − 3 ) − 3 = 3 − 3 = 0
y = ( − 3 ) 2 − 2 ( − 3 ) − 15 = 9 + 6 − 15 = 0 Since 0 = 0 , x = − 3 is a solution.


For x = 0 :

y = − ( 0 ) − 3 = − 3
y = ( 0 ) 2 − 2 ( 0 ) − 15 = − 15 Since − 3  = − 15 , x = 0 is not a solution.


For x = 4 :

y = − ( 4 ) − 3 = − 7
y = ( 4 ) 2 − 2 ( 4 ) − 15 = 16 − 8 − 15 = − 7 Since − 7 = − 7 , x = 4 is a solution.


Identifying the Solutions The solutions to the equation − x − 3 = x 2 − 2 x − 15 are the x-values for which the y-values of both equations are equal. From our calculations, these x-values are -3 and 4.


Examples
Understanding the intersection of graphs has practical applications in various fields. For instance, in economics, the intersection of supply and demand curves determines the market equilibrium price and quantity. Similarly, in physics, finding the intersection points of trajectories can help predict collisions. Graphing systems of equations and identifying their solutions is a fundamental skill that provides insights into real-world scenarios where multiple conditions must be satisfied simultaneously.

Answered by GinnyAnswer | 2025-07-08

The solutions to the equation − x − 3 = x 2 − 2 x − 15 are found at the points where the lines intersect. The values are − 3 and 4 . Therefore, the correct options are − 3 and 4 .
;

Answered by Anonymous | 2025-07-19