What substitution should be used to rewrite 4x⁴ - 21x² + 20 = 0 as a quadratic equation?
A. u = x²
B. u = 2x²
C. u = x⁴
D. u = 4x⁴
Answer (2)
Substitute u = x 2 into the equation.
Rewrite the equation in terms of u : 4 u 2 − 21 u + 20 = 0 .
The equation is now a quadratic equation in u .
The correct substitution is u = x 2 , so the answer is u = x 2 .
Explanation
Understanding the Problem We are given the equation 4 x 4 − 21 x 2 + 20 = 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 u is the variable.
Applying the Substitution Let's consider the substitution u = x 2 . Then, x 4 = ( x 2 ) 2 = u 2 . Substituting this into the given equation, we get:
4 ( x 2 ) 2 − 21 ( x 2 ) + 20 = 0
4 u 2 − 21 u + 20 = 0
This is a quadratic equation in u .
Conclusion Therefore, the correct substitution is u = x 2 .
Examples
Consider a scenario where you are analyzing the trajectory of a projectile. The equation 4 x 4 − 21 x 2 + 20 = 0 might arise when determining the points at which the projectile reaches a certain height. By using the substitution u = x 2 , you can transform this equation into a quadratic equation, which is much easier to solve. This allows you to find the values of x 2 , and subsequently, the values of x that correspond to the projectile's position at that height. This technique simplifies the analysis and provides a clear understanding of the projectile's motion.
To rewrite the equation 4 x 4 − 21 x 2 + 20 = 0 as a quadratic equation, use the substitution u = x 2 . This transforms the original equation into 4 u 2 − 21 u + 20 = 0 , which is a quadratic equation. Therefore, the correct choice is u = x 2 .
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