To determine the nature of the roots for the quadratic equation [tex]\(2x^2 + 5x - 7 = 0\)[/tex], we need to evaluate the discriminant of the equation. The discriminant ([tex]\(\Delta\)[/tex]) for a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
For the equation [tex]\(2x^2 + 5x - 7 = 0\)[/tex]:
- The coefficient [tex]\(a\)[/tex] is 2,
- The coefficient [tex]\(b\)[/tex] is 5,
- The constant term [tex]\(c\)[/tex] is -7.
Substituting these values into the discriminant formula:
[tex]\[
\Delta = 5^2 - 4 \cdot 2 \cdot (-7)
\][/tex]
Calculating this step by step:
1. [tex]\(5^2 = 25\)[/tex]
2. [tex]\(4 \cdot 2 = 8\)[/tex]
3. [tex]\(8 \cdot (-7) = -56\)[/tex]
4. Therefore, [tex]\(\Delta = 25 - (-56) = 25 + 56 = 81\)[/tex]
The discriminant [tex]\(\Delta\)[/tex] is 81.
Next, we interpret the value of the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real double root.
- If [tex]\(\Delta < 0\)[/tex], the equation has two non-real (complex) roots.
Since [tex]\(\Delta = 81\)[/tex] and [tex]\(81 > 0\)[/tex], we have two distinct real roots. To further classify these real roots as rational or irrational, we check if the discriminant is a perfect square:
- The square root of 81 is 9, which is an integer. Therefore, the discriminant is a perfect square.
Thus, the quadratic equation [tex]\(2x^2 + 5x - 7 = 0\)[/tex] has two real, rational roots.
The correct answer is:
A. Two real, rational roots
