To determine the nature of the roots for the quadratic equation [tex]\(2x^2 + 5x - 7 = 0\)[/tex], we can use the discriminant. The discriminant [tex]\(\Delta\)[/tex] for a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
For our equation, the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 5\)[/tex]
- [tex]\(c = -7\)[/tex]
First, let's calculate the discriminant:
[tex]\[
\Delta = 5^2 - 4 \cdot 2 \cdot (-7)
\][/tex]
[tex]\[
\Delta = 25 + 56
\][/tex]
[tex]\[
\Delta = 81
\][/tex]
With the discriminant calculated, we can determine the nature of the roots based on the value of [tex]\(\Delta\)[/tex]:
1. If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two distinct real roots.
- If [tex]\(\Delta\)[/tex] is a perfect square, those roots are rational.
- If [tex]\(\Delta\)[/tex] is not a perfect square, those roots are irrational.
2. If [tex]\(\Delta = 0\)[/tex], the quadratic equation has exactly one real double root.
3. If [tex]\(\Delta < 0\)[/tex], the quadratic equation has two non-real (complex) roots.
Here, our discriminant [tex]\(\Delta = 81\)[/tex] is positive and is also a perfect square (since [tex]\(81\)[/tex] is [tex]\(9^2\)[/tex]). Therefore, the quadratic equation [tex]\(2x^2 + 5x - 7 = 0\)[/tex] has two distinct real rational roots.
Thus, the correct answer is:
D. Two real, rational roots
