To determine the number of real roots for the quadratic equation [tex]\( y = -3x^2 + 12x - 12 \)[/tex], we can use the discriminant of the quadratic formula. The quadratic equation in standard form is given by:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Here, the coefficients are:
[tex]\[ a = -3 \][/tex]
[tex]\[ b = 12 \][/tex]
[tex]\[ c = -12 \][/tex]
The discriminant (Δ) of the quadratic equation is calculated using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 12^2 - 4(-3)(-12) \][/tex]
[tex]\[ \Delta = 144 - 144 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
The discriminant helps us determine the nature of the roots of the quadratic equation:
- If [tex]\(\Delta > 0\)[/tex], there are 2 distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], there is exactly 1 real root.
- If [tex]\(\Delta < 0\)[/tex], there are no real roots (the roots are complex).
In this case, the discriminant is [tex]\( \Delta = 0 \)[/tex], which indicates that there is exactly 1 real root.
Thus, the quadratic equation [tex]\( y = -3x^2 + 12x - 12 \)[/tex] has:
B. 1 real root.
