Solve $y^4-17 y^2+16=0$
Answer (1)
Substitute z = y 2 to transform the equation into a quadratic equation: z 2 − 17 z + 16 = 0 .
Factor the quadratic equation: ( z − 1 ) ( z − 16 ) = 0 , which gives z = 1 or z = 16 .
Substitute back y 2 = z and solve for y : y 2 = 1 gives y = ± 1 , and y 2 = 16 gives y = ± 4 .
The solutions are y = − 4 , − 1 , 1 , 4 , so the final answer is − 4 , − 1 , 1 , 4 .
Explanation
Problem Analysis We are given the equation y 4 − 17 y 2 + 16 = 0 . Our goal is to find all values of y that satisfy this equation.
Substitution To make the equation easier to solve, we can use a substitution. Let z = y 2 . Then the equation becomes z 2 − 17 z + 16 = 0 . This is a quadratic equation in terms of z .
Solving for z Now we solve the quadratic equation z 2 − 17 z + 16 = 0 for z . We can solve this by factoring. We look for two numbers that multiply to 16 and add to -17. These numbers are -1 and -16. So we can factor the quadratic as ( z − 1 ) ( z − 16 ) = 0 .
Possible Values of z From the factored equation ( z − 1 ) ( z − 16 ) = 0 , we have two possible values for z : z = 1 or z = 16 .
Solving for y Since z = y 2 , we substitute back to find the values of y . We have two cases:
Case 1: y 2 = 1 . Taking the square root of both sides, we get y = ± 1 .
Case 2: y 2 = 16 . Taking the square root of both sides, we get y = ± 4 .
Final Answer Therefore, the solutions to the equation y 4 − 17 y 2 + 16 = 0 are y = − 4 , − 1 , 1 , 4 .
Examples
Consider a scenario where you are designing a rectangular garden. The area of the garden is determined by the equation y 4 − 17 y 2 + 16 = 0 , where y represents a dimension related to the garden's sides. Solving this equation helps you find the possible values for that dimension, ensuring your garden design meets specific area requirements. Understanding how to solve such equations allows for precise planning and efficient use of space in real-world applications.
