To solve the quadratic equation [tex]\(4 - 4y - y^2 = 0\)[/tex] for [tex]\(y\)[/tex], follow these steps:
1. Identify the standard form of the quadratic equation:
The given quadratic equation is [tex]\(4 - 4y - y^2 = 0\)[/tex]. We can rewrite it as [tex]\(-y^2 - 4y + 4 = 0\)[/tex].
2. Rearrange the equation:
For convenience, we can multiply the entire equation by [tex]\(-1\)[/tex] to make it more standard:
[tex]\[
y^2 + 4y - 4 = 0.
\][/tex]
3. Solve the quadratic equation using the quadratic formula:
The quadratic formula is:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\][/tex]
For our equation [tex]\(y^2 + 4y - 4 = 0\)[/tex], the coefficients are:
[tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -4\)[/tex].
4. Calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[
\Delta = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-4) = 16 + 16 = 32.
\][/tex]
5. Substitute the values into the quadratic formula:
[tex]\[
y = \frac{-4 \pm \sqrt{32}}{2 \cdot 1} = \frac{-4 \pm 4\sqrt{2}}{2}.
\][/tex]
6. Simplify the solutions:
[tex]\[
y = \frac{-4 + 4\sqrt{2}}{2} = -2 + 2\sqrt{2},
\][/tex]
[tex]\[
y = \frac{-4 - 4\sqrt{2}}{2} = -2 - 2\sqrt{2}.
\][/tex]
Thus, the possible values of [tex]\(y\)[/tex] are:
[tex]\[
y = -2 + 2\sqrt{2}, \quad y = -2 - 2\sqrt{2}.
\][/tex]
Therefore, the correct answer is:
A. [tex]\(y = -2 + 2\sqrt{2}, y = -2 - 2\sqrt{2}\)[/tex].
