Certainly! To find the number of solutions (roots) for the quadratic equation [tex]\(y = 3x^2 - 8x + 8\)[/tex], we need to analyze its discriminant. A quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] has a discriminant given by the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
For the equation given:
[tex]\[
3x^2 - 8x + 8 = 0
\][/tex]
we identify the coefficients:
[tex]\[
a = 3, \quad b = -8, \quad c = 8
\][/tex]
Now we substitute these values into the discriminant formula:
[tex]\[
D = (-8)^2 - 4 \cdot 3 \cdot 8
\][/tex]
Simplifying this:
[tex]\[
D = 64 - 96
\][/tex]
[tex]\[
D = -32
\][/tex]
The value of the discriminant [tex]\(D\)[/tex] is [tex]\(-32\)[/tex]. The discriminant tells us about the nature and number of roots of the quadratic equation:
- If [tex]\(D > 0\)[/tex], there are 2 distinct real roots.
- If [tex]\(D = 0\)[/tex], there is exactly 1 real root (a repeated root).
- If [tex]\(D < 0\)[/tex], there are no real roots, only complex roots.
Since [tex]\(D = -32\)[/tex] is less than zero, this means that the quadratic equation [tex]\(3x^2 - 8x + 8 = 0\)[/tex] has no real roots.
Therefore, the number of real solutions for the equation [tex]\(y = 3x^2 - 8x + 8\)[/tex] is:
[tex]\[
\boxed{0}
\][/tex]
