(d) [tex]x^{1 / 2}-4 x^{1 / 4}+3=0[/tex]
[tex]x=[/tex]
Answer (1)
Substitute y = x 1/4 to transform the equation into a quadratic equation: y 2 − 4 y + 3 = 0 .
Factor the quadratic equation: ( y − 1 ) ( y − 3 ) = 0 , which gives y = 1 or y = 3 .
Substitute back to find x : x = y 4 , so x = 1 4 = 1 or x = 3 4 = 81 .
Verify the solutions: Both x = 1 and x = 81 satisfy the original equation, thus the final answer is 1 , 81 .
Explanation
Understanding the Problem We are given the equation x 1/2 − 4 x 1/4 + 3 = 0 and we need to find the value(s) of x that satisfy this equation. This equation involves fractional exponents of x , specifically x 1/2 and x 1/4 . We can rewrite x 1/2 as ( x 1/4 ) 2 . This allows us to transform the equation into a quadratic equation by using a suitable substitution.
Making a Substitution Let's make a substitution to simplify the equation. Let y = x 1/4 . Then, y 2 = ( x 1/4 ) 2 = x 1/2 . Substituting these into the original equation, we get: y 2 − 4 y + 3 = 0
Solving the Quadratic Equation Now we have a quadratic equation in terms of y . We can solve this equation by factoring. We are looking for two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3. So, we can factor the quadratic equation as follows: ( y − 1 ) ( y − 3 ) = 0 This gives us two possible solutions for y : y = 1 or y = 3
Finding the Values of x Since y = x 1/4 , we have x = y 4 . Now we substitute the values of y we found into this equation to find the corresponding values of x .If y = 1 , then x = 1 4 = 1 If y = 3 , then x = 3 4 = 81 So, we have two possible solutions for x : x = 1 and x = 81 .
Verifying the Solutions Now we need to verify these solutions by substituting them back into the original equation.For x = 1 : ( 1 ) 1/2 − 4 ( 1 ) 1/4 + 3 = 1 − 4 ( 1 ) + 3 = 1 − 4 + 3 = 0 So, x = 1 is a valid solution.For x = 81 : ( 81 ) 1/2 − 4 ( 81 ) 1/4 + 3 = 9 − 4 ( 3 ) + 3 = 9 − 12 + 3 = 0 So, x = 81 is also a valid solution.Therefore, the solutions to the equation x 1/2 − 4 x 1/4 + 3 = 0 are x = 1 and x = 81 .
Final Answer The solutions to the equation x 1/2 − 4 x 1/4 + 3 = 0 are x = 1 and x = 81 .
Examples
Imagine you are designing a garden and want to determine the area of a square plot where the length of the side is related to its area in a specific way, such as the equation we solved. Understanding how to solve equations with fractional exponents can help you calculate the exact dimensions needed to meet certain design criteria, ensuring your garden fits perfectly within the available space. This type of problem also appears in physics, when calculating the velocity of an object or the period of a pendulum.
