$\frac{x^4}{y^4}-\frac{7 x^2}{y^2}+1$
Answer (2)
To analyze y 4 x 4 − y 2 7 x 2 + 1 , we substitute z = y 2 x 2 to rewrite it as a quadratic z 2 − 7 z + 1 . We then find the roots using the quadratic formula, approximate them, and analyze the behavior of the expression based on these roots. The parabola opens upwards, indicating the expression is negative between the roots and positive elsewhere.
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Substitute z = y 2 x 2 to rewrite the expression as z 2 − 7 z + 1 .
Find the roots of the quadratic equation z 2 − 7 z + 1 = 0 using the quadratic formula.
The roots are z = 2 7 ± 3 5 , approximately 0.146 and 6.854 .
Analyze the quadratic expression based on its roots and parabolic nature. The expression is negative between the roots and positive outside the roots.
Explanation
Simplifying the Expression We are given the expression y 4 x 4 − y 2 7 x 2 + 1 and we want to analyze it. Let's make a substitution to simplify the expression.
Substitution Let z = y 2 x 2 . Then the expression becomes z 2 − 7 z + 1 . This is a quadratic expression in terms of z .
Finding the Roots To further analyze this quadratic expression, we can find its roots by setting it equal to zero: z 2 − 7 z + 1 = 0 . We can use the quadratic formula to find the roots: z = 2 a − b ± b 2 − 4 a c In this case, a = 1 , b = − 7 , and c = 1 .
Calculating the Roots Plugging in the values, we get: z = 2 ( 1 ) 7 ± ( − 7 ) 2 − 4 ( 1 ) ( 1 ) = 2 7 ± 49 − 4 = 2 7 ± 45 = 2 7 ± 3 5 So the roots are z 1 = 2 7 − 3 5 and z 2 = 2 7 + 3 5 .
Approximating the Roots The roots are real and distinct. We can approximate these values to get a better understanding of their magnitude. z 1 = 2 7 − 3 5 ≈ 2 7 − 3 ( 2.236 ) ≈ 2 7 − 6.708 ≈ 2 0.292 ≈ 0.146 z 2 = 2 7 + 3 5 ≈ 2 7 + 3 ( 2.236 ) ≈ 2 7 + 6.708 ≈ 2 13.708 ≈ 6.854
Analyzing the Expression Since z = y 2 x 2 , we have y 2 x 2 = 2 7 ± 3 5 . This means that y x = ± 2 7 ± 3 5 . The expression z 2 − 7 z + 1 is a parabola opening upwards. It is negative between the roots and positive outside the roots.
Examples
Understanding quadratic expressions like this is crucial in many areas, such as physics, where projectile motion can be modeled using quadratic equations. For example, if you were designing a catapult, you might use a quadratic expression to model the trajectory of the projectile. By analyzing the roots and the shape of the parabola, you can determine the range and maximum height of the projectile, allowing you to optimize your design for maximum performance.
