To determine the number of real roots of the quadratic equation [tex]\( y = -2x^2 - 4x - 6 \)[/tex], we need to analyze its discriminant. The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Here, the coefficients from the quadratic equation [tex]\( y = -2x^2 - 4x - 6 \)[/tex] are:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = -6 \)[/tex]
Plugging these values into the discriminant formula, we get:
[tex]\[ \Delta = (-4)^2 - 4(-2)(-6) \][/tex]
Calculating each part step-by-step:
- [tex]\( (-4)^2 = 16 \)[/tex]
- [tex]\( 4 \times (-2) \times (-6) = 48 \)[/tex]
(since [tex]\( 4 \times (-2) = -8 \)[/tex] and [tex]\( -8 \times (-6) = 48 \)[/tex])
So, the discriminant is:
[tex]\[ \Delta = 16 - 48 \][/tex]
[tex]\[ \Delta = -32 \][/tex]
The sign of the discriminant helps us determine the number of real roots of the quadratic equation:
- If [tex]\(\Delta > 0\)[/tex], the equation has 2 distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly 1 real root.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real roots (the roots are complex).
In our case, [tex]\(\Delta = -32 \)[/tex], which is less than 0. Therefore, the quadratic equation [tex]\( y = -2x^2 - 4x - 6 \)[/tex] has no real roots.
The answer is:
A. 0
