To determine the nature of the roots of the quadratic equation [tex]\( x^2 + 7x - 60 = 0 \)[/tex], we need to evaluate the 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]
For the equation [tex]\( x^2 + 7x - 60 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = -60 \)[/tex]
Now, we substitute these values into the discriminant formula to find [tex]\(\Delta\)[/tex]:
[tex]\[ \Delta = 7^2 - 4 \cdot 1 \cdot (-60) \][/tex]
[tex]\[ \Delta = 49 + 240 \][/tex]
[tex]\[ \Delta = 289 \][/tex]
Given that the discriminant [tex]\(\Delta\)[/tex] is 289, we analyze the nature of the roots based on the value of the discriminant:
1. If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two real and unequal roots.
2. If [tex]\(\Delta = 0\)[/tex], the quadratic equation has two equal (real) roots.
3. If [tex]\(\Delta < 0\)[/tex], the quadratic equation has no real roots (the roots are complex and conjugate pairs).
Since [tex]\(\Delta = 289\)[/tex] which is greater than 0, the quadratic equation [tex]\( x^2 + 7x - 60 = 0 \)[/tex] has two real and unequal roots.
Therefore, the correct answer is:
B. two real and unequal roots
