Let's solve the problem step-by-step.
### 1. Find the Discriminant
The given quadratic equation is:
[tex]\[ 2x^2 - 5x + 7 = 0 \][/tex]
The general form of a quadratic equation is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
where [tex]\( a = 2 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 7 \)[/tex].
The discriminant ([tex]\( \Delta \)[/tex]) of a quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the given values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[ \Delta = (-5)^2 - 4 \cdot 2 \cdot 7 \][/tex]
Calculate each term:
[tex]\[ \Delta = 25 - 4 \cdot 2 \cdot 7 \][/tex]
[tex]\[ \Delta = 25 - 56 \][/tex]
[tex]\[ \Delta = -31 \][/tex]
So, the discriminant is [tex]\( \Delta = -31 \)[/tex].
### 2. Describe the Roots and Explain Your Reasoning
The value of the discriminant ([tex]\( \Delta \)[/tex]) helps in determining the nature of the roots of the quadratic equation. There are three cases to consider:
1. If [tex]\( \Delta > 0 \)[/tex]: The quadratic equation has two distinct real roots.
2. If [tex]\( \Delta = 0 \)[/tex]: The quadratic equation has exactly one real root (a repeated or double root).
3. If [tex]\( \Delta < 0 \)[/tex]: The quadratic equation has no real roots; instead, it has two complex (conjugate) roots.
In this case, we found [tex]\( \Delta = -31 \)[/tex], which is less than zero ([tex]\( \Delta < 0 \)[/tex]). Therefore, the quadratic equation does not have any real roots. Instead, it has two complex (conjugate) roots.
Thus, the roots of the equation [tex]\( 2x^2 - 5x + 7 = 0 \)[/tex] are complex. The description of the roots is:
"The equation has no real roots (complex roots)."
