Question

The difference between the maximum real root and the minimum real root of the equation \( (x^2 - 5)^4 + (x^2 - 7)^4 = 16 \) is:

• A. \( 2\sqrt{5} \)
• B. \( 2\sqrt{7} \)
• C. \( \sqrt{7} \)
• D. \( \sqrt{10} \)

Hint:

For quartic equations involving sum of squares, trial and error or graphing methods can sometimes help find the roots more efficiently.

Answer 1

The Correct Option is B

Step 1: Let \( y = x^2 \). We substitute \( x^2 = y \) to reduce the equation to a more manageable form. The equation becomes: \[ (y - 5)^4 + (y - 7)^4 = 16. \] This equation is a quartic in \( y \), which can be solved by substitution or numerical methods. Step 2: Trial and error or numerical methods To solve this quartic equation, we can use trial and error, factoring, or numerical methods. After applying these methods, we find the two real roots for \( y \). Step 3: Take square roots to find the real roots for \( x \) Once we have the roots for \( y \), we take the square roots to find the corresponding real roots for \( x \). The two values for \( x \) give us the maximum and minimum real roots. Step 4: Find the difference between the maximum and minimum real roots After calculating the roots of \( x \), we find that the difference between the maximum and minimum real roots is \( 2\sqrt{7} \). Thus, the correct answer is \rupee\rupee(B)\rupee\rupee \( 2\sqrt{7} \).