To solve the quadratic equation [tex]\(-1 = 5x^2 - 2x\)[/tex] and determine the value of the discriminant, as well as what the discriminant value signifies about the number of real solutions, follow these steps:
1. Rewrite the Equation in Standard Form:
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. Rewrite the given equation in this form:
[tex]\[
5x^2 - 2x + 1 = 0
\][/tex]
2. Identify the Coefficients:
Here, the coefficients are:
[tex]\[
a = 5, \quad b = -2, \quad c = 1
\][/tex]
3. Calculate the Discriminant:
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[
D = (-2)^2 - 4(5)(1) = 4 - 20 = -16
\][/tex]
4. Interpret the Discriminant:
- If the discriminant [tex]\(D > 0\)[/tex], the quadratic equation has two distinct real number solutions.
- If [tex]\(D = 0\)[/tex], the quadratic equation has exactly one real number solution.
- If [tex]\(D < 0\)[/tex], the quadratic equation has no real number solutions, but rather two complex solutions.
Since the discriminant is [tex]\(-16\)[/tex] which is less than zero, the quadratic equation [tex]\(5x^2 - 2x + 1 = 0\)[/tex] has no real number solutions.
Thus, the correct interpretation is:
- The discriminant is equal to [tex]\(-16\)[/tex], which means the equation has no real number solutions.
