To determine the number of real roots for the equation [tex]\( y = 2(x + 1)^2 \)[/tex], we can analyze the properties of this quadratic equation step-by-step:
1. Identify the equation form: The equation given is [tex]\( y = 2(x + 1)^2 \)[/tex]. This is a quadratic equation in the form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( h = -1 \)[/tex], and [tex]\( k = 0 \)[/tex].
2. Shape of the graph: Since the coefficient [tex]\( a = 2 \)[/tex] is positive, the parabola opens upwards. In general, if [tex]\( a > 0 \)[/tex], the parabola is convex and opens upwards; if [tex]\( a < 0 \)[/tex], it opens downwards. Here, it opens upwards.
3. Vertex of the parabola: The vertex of the parabola [tex]\( y = a(x - h)^2 + k \)[/tex] is at the point [tex]\((h, k)\)[/tex]. For our equation, the vertex is at [tex]\((-1, 0)\)[/tex]. This vertex point is also the minimum point on the graph of the parabola.
4. Minimum value of the function: At the vertex [tex]\((x, y) = (-1, 0)\)[/tex], the value of [tex]\( y \)[/tex] is zero. Since the parabola opens upwards, the function only touches the x-axis at this single point.
5. Solving for [tex]\( x \)[/tex] at [tex]\( y = 0 \)[/tex]:
[tex]\[
0 = 2(x + 1)^2
\][/tex]
To find the values of [tex]\( x \)[/tex] that satisfy this equation, divide both sides by 2:
[tex]\[
0 = (x + 1)^2
\][/tex]
Solving the above equation, we set the expression inside the square to zero:
[tex]\[
(x + 1)^2 = 0
\][/tex]
Take the square root of both sides:
[tex]\[
x + 1 = 0
\][/tex]
Finally, solve for [tex]\( x \)[/tex]:
[tex]\[
x = -1
\][/tex]
6. Conclusion: We have determined that the equation [tex]\( y = 2(x + 1)^2 \)[/tex] has precisely one real solution for [tex]\( x \)[/tex], which is [tex]\( x = -1 \)[/tex].
Hence, the number of real roots of the equation [tex]\( y = 2(x + 1)^2 \)[/tex] is 1. The correct answer is:
A. 1
