Find all real solutions. (Enter your answers as comma-separated lists. If there is no real solution, enter NO REALSOLUTION.)
[tex]x^4-7 x^2+6=0[/tex]
[tex]x=\text { ? }[/tex]
Answer (1)
Substitute y = x 2 to transform the equation into a quadratic equation: y 2 − 7 y + 6 = 0 .
Factor the quadratic equation to find the solutions for y : y = 1 and y = 6 .
Substitute back x = ± y to find the solutions for x : x = ± 1 and x = ± 6 .
The real solutions are x = − 6 , − 1 , 1 , 6 , so the final answer is − 6 , − 1 , 1 , 6 .
Explanation
Understanding the Problem We are given the equation x 4 − 7 x 2 + 6 = 0 and asked to find all real solutions for x .
Substitution Let y = x 2 . Then the equation becomes y 2 − 7 y + 6 = 0 . This is a quadratic equation in terms of y .
Solving for y We can factor the quadratic equation as ( y − 6 ) ( y − 1 ) = 0 . Therefore, the solutions for y are y = 6 and y = 1 .
Solving for x Since y = x 2 , we have x = ± y . For y = 6 , we have x = ± 6 . For y = 1 , we have x = ± 1 = ± 1 .
Finding the Solutions Thus, the real solutions for x are 6 , − 6 , 1 , and − 1 .
Final Answer Therefore, the real solutions are x = − 6 , − 1 , 1 , 6 .
Examples
Consider a landscape design where the area is modeled by a quartic equation. Finding the real solutions helps determine the actual, physically possible dimensions of the landscape. For instance, if x represents a length, only real solutions make sense. This type of problem arises in various engineering and design contexts where polynomial equations model physical quantities.
