Find the solution(s) of the system of equations:
[tex]
\begin{array}{l}
y=-x^2+4 x+5 \\
y=x^2+2 x+1
\end{array}
[/tex]
Answer (1)
Set the two equations equal to each other: − x 2 + 4 x + 5 = x 2 + 2 x + 1 .
Rearrange to form a quadratic equation: 2 x 2 − 2 x − 4 = 0 , which simplifies to x 2 − x − 2 = 0 .
Factor the quadratic equation: ( x − 2 ) ( x + 1 ) = 0 , so x = 2 or x = − 1 .
Substitute x values to find corresponding y values: solutions are ( − 1 , 0 ) and ( 2 , 9 ) .
Explanation
Understanding the Problem We are given a system of two equations:
y = − x 2 + 4 x + 5 and y = x 2 + 2 x + 1 .
The objective is to find the solution(s) of this system of equations.
Setting the Equations Equal To find the solutions, we need to find the points ( x , y ) that satisfy both equations simultaneously. This means we need to solve the system of equations.
We can set the two equations equal to each other: − x 2 + 4 x + 5 = x 2 + 2 x + 1
Now, we rearrange the equation to form a quadratic equation in x : 2 x 2 − 2 x − 4 = 0
We simplify the quadratic equation by dividing by 2: x 2 − x − 2 = 0
Factoring the Quadratic Equation Now, we factor the quadratic equation: ( x − 2 ) ( x + 1 ) = 0
Solve for x :
x = 2 or x = − 1
Finding the Corresponding y Values Substitute each value of x into either of the original equations to find the corresponding y values.
If x = 2 , then y = ( 2 ) 2 + 2 ( 2 ) + 1 = 4 + 4 + 1 = 9 So one solution is ( 2 , 9 ) .
If x = − 1 , then y = ( − 1 ) 2 + 2 ( − 1 ) + 1 = 1 − 2 + 1 = 0 So another solution is ( − 1 , 0 ) .
Therefore, the solutions are ( − 1 , 0 ) and ( 2 , 9 ) .
Final Answer The solutions to the system of equations are ( − 1 , 0 ) and ( 2 , 9 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, modeling supply and demand in economics, and analyzing the trajectory of objects in physics. In this case, the intersection points of two parabolas represent the solutions to the system, which can be visualized as the points where the graphs of the two equations meet. Understanding how to solve systems of equations is crucial for making informed decisions and solving complex problems in many fields.
