Approximate the three solutions to the following equation:
[tex]x^4-\frac{11}{5} x^3-1=\frac{1}{x}-3[/tex]
These are the three solutions in order from least to greatest:
[tex]\begin{array}{l}
x =\square \
x=\square \
x=\square
\end{array}[/tex]
Answer (2)
Rewrite the given equation as x 5 − 5 11 x 4 + 2 x − 1 = 0 .
Define the function f ( x ) = x 5 − 5 11 x 4 + 2 x − 1 .
Find the approximate roots of f ( x ) = 0 using a numerical method.
Order the roots from least to greatest: 0.6102 , 0.8439 , 2.0166 .
Explanation
Problem Analysis We are given the equation x 4 − 5 11 x 3 − 1 = x 1 − 3 . Our goal is to approximate the three real solutions to this equation and order them from least to greatest.
Rewriting the Equation First, let's rewrite the equation to have all terms on one side. We can do this by adding 3 to both sides and subtracting x 1 from both sides, which gives us x 4 − 5 11 x 3 + 2 − x 1 = 0 . To get rid of the fraction, we multiply the entire equation by x , resulting in x 5 − 5 11 x 4 + 2 x − 1 = 0 .
Defining the Function Let f ( x ) = x 5 − 5 11 x 4 + 2 x − 1 . We want to find the roots of this polynomial. We can use numerical methods or graphing tools to approximate the roots. By using a numerical root-finding method, we find three real roots.
Finding the Roots Using a numerical method, we find the approximate roots of the equation f ( x ) = x 5 − 5 11 x 4 + 2 x − 1 = 0 . The three real roots are approximately 0.6102 , 0.8439 , and 2.0166 .
Ordering the Roots Ordering these roots from least to greatest, we have x ≈ 0.6102 , x ≈ 0.8439 , and x ≈ 2.0166 .
Final Answer Therefore, the three solutions, ordered from least to greatest, are approximately 0.6102 , 0.8439 , and 2.0166 .
Examples
Approximating solutions to equations like this is useful in many fields, such as engineering and physics, where exact solutions may not be possible to find analytically. For example, when designing a bridge, engineers might need to solve complex equations to determine the optimal dimensions and materials to use. Similarly, in physics, approximating solutions to equations can help scientists understand the behavior of complex systems, such as the motion of planets or the flow of fluids. These approximations allow for practical applications and informed decision-making in real-world scenarios.
The solutions to the equation are approximately 0.6102, 0.8439, and 2.0166. These roots are determined by rewriting the equation and using numerical methods to approximate their values. The final ordered list of solutions is from least to greatest: 0.6102, 0.8439, and 2.0166.
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