[tex]x^4-3 x^2+2=0[/tex]
Answer (2)
Substitute y = x 2 to transform the equation into a quadratic equation in y .
Factor the quadratic equation to find the values of y : y = 1 and y = 2 .
Substitute back x 2 for y and solve for x .
The solutions are x = ± 1 and x = ± 2 , so x = − 2 , − 1 , 1 , 2 .
Explanation
Understanding the Problem We are given the equation x 4 − 3 x 2 + 2 = 0 and asked to find the values of x that satisfy this equation. This is a quartic equation, but we can solve it by recognizing that it is quadratic in x 2 .
Making a Substitution Let's make a substitution to simplify the equation. Let y = x 2 . Then the equation becomes y 2 − 3 y + 2 = 0 .
Factoring the Quadratic Equation Now we can factor the quadratic equation in y : y 2 − 3 y + 2 = ( y − 1 ) ( y − 2 ) = 0 .
Solving for y Solving for y , we have two possible values: y = 1 or y = 2 .
Substituting Back and Solving for x Now we substitute back x 2 for y to find the values of x .
If y = 1 , then x 2 = 1 , which means x = ± 1 .
If y = 2 , then x 2 = 2 , which means x = ± 2 .
Final Answer Therefore, the solutions for x are x = − 2 , − 1 , 1 , 2 .
Examples
Consider a landscape architect designing a park with a symmetrical layout. The equation x 4 − 3 x 2 + 2 = 0 could represent the boundaries or key points of interest within the park. Solving this equation helps the architect determine the exact locations for these features, ensuring a balanced and aesthetically pleasing design. Understanding the roots of such equations allows for precise planning and execution in creating harmonious outdoor spaces.
To solve the equation x 4 − 3 x 2 + 2 = 0 , we substitute y = x 2 to obtain a quadratic equation that factors to yield values for y . By substituting back to find x , we find the solutions are x = − 2 , − 1 , 1 , 2 .
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