To determine the value of the discriminant for the quadratic equation and interpret its meaning regarding the number of real solutions, follow these steps:
1. Rewrite the equation in standard form:
The given equation is:
[tex]\[
-2x^2 = -8x + 8
\][/tex]
First, move all terms to one side of the equation to set it equal to zero:
[tex]\[
-2x^2 + 8x - 8 = 0
\][/tex]
Now, the equation is in standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
[tex]\[
a = -2, \quad b = 8, \quad c = -8
\][/tex]
2. Calculate 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]
Plug in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 8^2 - 4(-2)(-8)
\][/tex]
[tex]\[
\Delta = 64 - 4 \cdot (-2) \cdot (-8)
\][/tex]
[tex]\[
\Delta = 64 - 64
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
3. Interpret the value of the discriminant:
The discriminant value [tex]\(\Delta\)[/tex] determines the nature and number of real solutions of the quadratic equation:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real solution.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real solutions.
Since the discriminant [tex]\(\Delta = 0\)[/tex], this indicates that the equation has exactly one real solution.
Therefore, the correct interpretation is:
- The discriminant is equal to 0, which means the equation has one real number solution.
