To find the value or values of [tex]\( y \)[/tex] in the quadratic equation [tex]\( y^2 + 4y + 4 = 7 \)[/tex], we can follow these steps:
1. Rewrite the equation:
Start by subtracting 7 from both sides of the equation to set it to zero:
[tex]\[
y^2 + 4y + 4 - 7 = 0
\][/tex]
Simplify this to:
[tex]\[
y^2 + 4y - 3 = 0
\][/tex]
So now, we need to solve the quadratic equation:
[tex]\[
y^2 + 4y - 3 = 0
\][/tex]
2. Identify coefficients:
In the standard form of a quadratic equation [tex]\( ay^2 + by + c = 0 \)[/tex],
[tex]\[
a = 1, \quad b = 4, \quad c = -3
\][/tex]
3. Calculate the discriminant:
The discriminant of a quadratic equation is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Plugging in the values, we get:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot (-3)
\][/tex]
Simplifying this:
[tex]\[
\Delta = 16 + 12 = 28
\][/tex]
4. Solve for [tex]\( y \)[/tex] using the quadratic formula:
The quadratic formula is [tex]\( y = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex].
Plug in the values we identified:
[tex]\[
y = \frac{-4 \pm \sqrt{28}}{2 \cdot 1}
\][/tex]
Simplify the expression:
[tex]\[
y = \frac{-4 \pm \sqrt{28}}{2}
\][/tex]
5. Simplify the square root:
Since [tex]\( \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \)[/tex],
[tex]\[
y = \frac{-4 \pm 2\sqrt{7}}{2}
\][/tex]
6. Further simplify:
Split and simplify the fraction:
[tex]\[
y = \frac{-4}{2} \pm \frac{2\sqrt{7}}{2}
\][/tex]
[tex]\[
y = -2 \pm \sqrt{7}
\][/tex]
7. Find the two solutions:
Therefore, the solutions for [tex]\( y \)[/tex] are:
[tex]\[
y = -2 + \sqrt{7} \approx 0.6457513110645907
\][/tex]
and
[tex]\[
y = -2 - \sqrt{7} \approx -4.645751311064591
\][/tex]
So the values of [tex]\( y \)[/tex] that satisfy the original equation are approximately [tex]\( \boxed{0.6457513110645907 \text{ and } -4.645751311064591} \)[/tex].
