Find the solution to the system of equations. Round to the nearest tenth if necessary.
[tex]
\begin{array}{l}
f(x)=\sqrt{2 x+4} \\
g(x)=x^3-4
\end{array}
[/tex]
Answer (1)
Set the two functions equal to each other: 2 x + 4 = x 3 − 4 .
Square both sides and rearrange to get a polynomial equation: x 6 − 8 x 3 − 2 x + 12 = 0 .
Find the approximate real roots using numerical methods: x ≈ 1.14 and x ≈ 1.89 .
Check for extraneous solutions and round to the nearest tenth: 1.9 .
Explanation
Problem Analysis We are given two functions, f ( x ) = \t \t 2 x + 4 and g ( x ) = x 3 − 4 , and we want to find the solutions to the system of equations f ( x ) = g ( x ) . This means we need to find the values of x for which the two functions are equal.
Setting up the Equation To solve this system, we set the two functions equal to each other: \t\t 2 x + 4 = x 3 − 4
\t\tTo eliminate the square root, we square both sides of the equation: \t\t ( 2 x + 4 ) 2 = ( x 3 − 4 ) 2
\t\tThis simplifies to: \t\t 2 x + 4 = x 6 − 8 x 3 + 16
Forming a Polynomial Equation Now, we rearrange the equation to form a polynomial equation: \t\t x 6 − 8 x 3 − 2 x + 12 = 0
\t\tThis is a sixth-degree polynomial equation, which is difficult to solve analytically. Therefore, we will use numerical methods or a graphing calculator to find approximate solutions.
Finding Approximate Roots Using a numerical method, we find the approximate real roots of the equation x 6 − 8 x 3 − 2 x + 12 = 0 to be approximately x \t \t ≈ 1.14 and x \t \t ≈ 1.89 . We need to check these solutions in the original equation to make sure they are valid, since squaring both sides can introduce extraneous solutions.
Checking for Extraneous Solutions Let's check x \t \t ≈ 1.14 : \t\t f ( 1.14 ) = 2 ( 1.14 ) + 4 = 2.28 + 4 = 6.28 \t \t ≈ 2.51
\t\t g ( 1.14 ) = ( 1.14 ) 3 − 4 = 1.48 − 4 = − 2.52
\t\tSince f ( 1.14 ) \t \t = g ( 1.14 ) , x \t \t ≈ 1.14 is not a solution.
\t\tNow let's check x \t \t ≈ 1.89 : \t\t f ( 1.89 ) = 2 ( 1.89 ) + 4 = 3.78 + 4 = 7.78 \t \t ≈ 2.79
\t\t g ( 1.89 ) = ( 1.89 ) 3 − 4 = 6.76 − 4 = 2.76
\t\tSince f ( 1.89 ) \t \t ≈ g ( 1.89 ) , x \t \t ≈ 1.89 is a valid solution.
Final Solution Therefore, the solution to the system of equations, rounded to the nearest tenth, is x \t \t ≈ 1.9 .
Examples
Systems of equations are used in many real-world applications, such as determining the intersection points of two paths, optimizing resource allocation, or modeling the behavior of physical systems. For example, if you have a drone flying along a path defined by f ( x ) and another drone flying along a path defined by g ( x ) , finding the solution to the system of equations f ( x ) = g ( x ) will tell you where the paths intersect, helping you avoid collisions or coordinate their movements.
