What substitution should be used to rewrite $4 x^{12}-5 x^6-14=0$ as a quadratic equation?
A. $u=x^2$
B. $u=x^3$
C. $u=x^6$
D. $u=x^{12}$
Answer (2)
Recognize that the given equation 4 x 12 − 5 x 6 − 14 = 0 can be transformed into a quadratic equation by using an appropriate substitution.
Test each of the given options to see which one results in a quadratic equation.
Substitute u = x 6 into the equation, which yields 4 u 2 − 5 u − 14 = 0 , a quadratic equation.
Conclude that the correct substitution is u = x 6 .
Explanation
Understanding the Problem We are given the equation 4 x 12 − 5 x 6 − 14 = 0 and asked to find a substitution that transforms it into a quadratic equation. A quadratic equation has the form a u 2 + b u + c = 0 , where a , b , and c are constants and u is a variable. We need to find a substitution u = x n such that the given equation can be written in this form.
Testing the Options Let's test the given options:
If u = x 2 , then x 12 = ( x 2 ) 6 = u 6 and x 6 = ( x 2 ) 3 = u 3 . Substituting into the equation gives 4 u 6 − 5 u 3 − 14 = 0 , which is not a quadratic equation.
If u = x 3 , then x 12 = ( x 3 ) 4 = u 4 and x 6 = ( x 3 ) 2 = u 2 . Substituting into the equation gives 4 u 4 − 5 u 2 − 14 = 0 , which is not a quadratic equation.
If u = x 6 , then x 12 = ( x 6 ) 2 = u 2 . Substituting into the equation gives 4 u 2 − 5 u − 14 = 0 , which is a quadratic equation.
If u = x 12 , then x 12 = u . Substituting into the equation gives 4 u − 5 u − 14 = 0 , which is not a quadratic equation.
Conclusion The substitution u = x 6 transforms the given equation into the quadratic equation 4 u 2 − 5 u − 14 = 0 . Therefore, the correct substitution is u = x 6 .
Examples
In physics, equations involving high powers can sometimes be simplified using substitutions to make them easier to solve. For example, when analyzing the motion of objects under certain forces, you might encounter equations similar to the one in this problem. By using a suitable substitution, you can reduce the complexity of the equation and find a solution more easily. This technique is also useful in engineering when dealing with polynomial equations that model system behavior.
The correct substitution to rewrite the equation 4 x 12 − 5 x 6 − 14 = 0 as a quadratic equation is u = x 6 . This leads to the quadratic form 4 u 2 − 5 u − 14 = 0 . Therefore, the answer is option C.
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