Solve the system of equations. Type in all points of intersection for the two functions and round to the nearest tenth if necessary.
[tex]
\begin{array}{l}
f(x)=\sqrt{5 x+25}-1 \\
g(x)=x^2-2 x-3
\end{array}
[/tex]
Answer (2)
Set the two functions equal: 5 x + 25 − 1 = x 2 − 2 x − 3 .
Isolate the square root and square both sides: 5 x + 25 = ( x 2 − 2 x − 2 ) 2 .
Simplify to get a quartic equation: x 4 − 4 x 3 + 3 x − 21 = 0 .
Find the real roots and corresponding y-values, then round to the nearest tenth: ( − 1.7 , 3.1 ) , ( 4.1 , 5.8 ) .
Explanation
Problem Analysis We are given two functions, f ( x ) = 5 x + 25 − 1 and g ( x ) = x 2 − 2 x − 3 , and we want to find their points of intersection. This means we need to solve the equation f ( x ) = g ( x ) for x .
Equating the Functions Setting the two functions equal to each other, we have 5 x + 25 − 1 = x 2 − 2 x − 3.
Isolating the Square Root Isolating the square root, we get 5 x + 25 = x 2 − 2 x − 2.
Squaring Both Sides Squaring both sides, we obtain 5 x + 25 = ( x 2 − 2 x − 2 ) 2 = x 4 − 4 x 3 + 4 x + 4.
Rearranging to Quartic Equation Rearranging the terms, we get the quartic equation x 4 − 4 x 3 − 4 x + 8 x − 21 = 0.
Simplifying the Equation Simplifying, we have x 4 − 4 x 3 + 3 x − 21 = 0.
Finding the Roots Using a calculator, we find the approximate real roots of this equation to be x ≈ − 1.6618 and x ≈ 4.1231 .
Finding the Corresponding y-values Now we need to find the corresponding y -values for these x -values. We can use either f ( x ) or g ( x ) . Let's use f ( x ) = 5 x + 25 − 1 .
For x ≈ − 1.6618 , we have f ( − 1.6618 ) = 5 ( − 1.6618 ) + 25 − 1 ≈ 16.691 − 1 ≈ 4.0854 − 1 ≈ 3.0854. So, y ≈ 3.0854 .
For x ≈ 4.1231 , we have f ( 4.1231 ) = 5 ( 4.1231 ) + 25 − 1 ≈ 45.6155 − 1 ≈ 6.7539 − 1 ≈ 5.7539. So, y ≈ 5.7539 .
Rounding the Coordinates Rounding the coordinates to the nearest tenth, we get the points of intersection as ( − 1.7 , 3.1 ) and ( 4.1 , 5.8 ) .
Final Answer Therefore, the points of intersection are approximately ( − 1.7 , 3.1 ) and ( 4.1 , 5.8 ) .
Examples
Understanding the intersection of functions is crucial in many real-world applications. For instance, in economics, the intersection of supply and demand curves determines the market equilibrium point, indicating the price and quantity at which the market clears. Similarly, in physics, finding the intersection points of motion equations can help determine when and where two objects will collide. These concepts are fundamental in making informed decisions and predictions in various fields.
The points of intersection for the functions are approximately (-1.7, 3.1) and (4.1, 5.8). The process involved setting the functions equal, isolating the square root, squaring both sides, and solving the resulting quartic equation. Rounding the roots to the nearest tenth gives the final points of intersection.
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