Certainly! To determine the number of real number solutions for the quadratic equation [tex]\( x^2 + 5x + 7 = 0 \)[/tex], we can use the discriminant. The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the given equation [tex]\( x^2 + 5x + 7 = 0 \)[/tex]:
- The coefficient [tex]\( a = 1 \)[/tex]
- The coefficient [tex]\( b = 5 \)[/tex]
- The constant term [tex]\( c = 7 \)[/tex]
Substitute these values into the discriminant formula:
[tex]\[ \Delta = 5^2 - 4 \cdot 1 \cdot 7 \][/tex]
[tex]\[ \Delta = 25 - 28 \][/tex]
[tex]\[ \Delta = -3 \][/tex]
The value of the discriminant [tex]\(\Delta\)[/tex] is [tex]\(-3\)[/tex].
The rules for the discriminant in determining the number of real solutions are as follows:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], there is exactly one real root.
- If [tex]\(\Delta < 0\)[/tex], there are no real roots (the roots are complex numbers).
Since [tex]\(\Delta = -3\)[/tex] is less than 0, the quadratic equation [tex]\( x^2 + 5x + 7 = 0 \)[/tex] has no real number solutions.
Therefore, the number of real number solutions for the given equation is:
[tex]\[ \boxed{0} \][/tex]
