Question

find two functions defined implicitly by this equation.
$4x^2 + y^2 = 25$
$y = \pm\sqrt{?} + \quadx^2$

Explanation:

Step1: Isolate \( y^2 \)

Start with the equation \( 4x^2 + y^2 = 25 \). Subtract \( 4x^2 \) from both sides to get \( y^2 = 25 - 4x^2 \).

Step2: Solve for \( y \)

Take the square root of both sides. Remember that when taking the square root, we consider both the positive and negative roots. So \( y = \pm\sqrt{25 - 4x^2} \). Comparing this to the given form \( y = \pm\sqrt{[?] + [\square]x^2} \), we can rewrite \( 25 - 4x^2 \) as \( 25 + (-4)x^2 \).

Answer:

The green box (the constant term) should be \( 25 \) and the gray box (the coefficient of \( x^2 \)) should be \( -4 \). So filling in the blanks, we have \( y = \pm\sqrt{25 + (-4)x^2} \), meaning the green box is \( 25 \) and the gray box is \( -4 \).