Question

what is the value of the discriminant of the quadratic equation -1 = 5x² - 2x, and what does its value mean about the number of real number solutions the equation has?
the discriminant is equal to -16, which means the equation has no real number solutions.
the discriminant is equal to -16, which means the equation has two real number solutions.
the discriminant is equal to 24, which means the equation has no real number solutions.
the discriminant is equal to 24, which means the equation has two real number solutions.

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the given equation \(-1 = 5x^2 - 2x\) in the standard quadratic form \(ax^2 + bx + c = 0\). By adding \(1\) to both sides and rearranging, we get \(5x^2 - 2x + 1 = 0\). Here, \(a = 5\), \(b = -2\), and \(c = 1\).

Step2: Calculate the discriminant

The formula for the discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is \(\Delta = b^2 - 4ac\). Substituting the values of \(a\), \(b\), and \(c\) we found:
\[

$$\begin{align*} \Delta&= (-2)^2 - 4\times5\times1\\ &= 4 - 20\\ &= -16 \end{align*}$$

\]

Step3: Analyze the discriminant

The discriminant determines the number of real solutions:

• If \(\Delta>0\), there are two distinct real solutions.
• If \(\Delta = 0\), there is one real solution (a repeated root).
• If \(\Delta<0\), there are no real solutions (two complex solutions).

Since \(\Delta=-16<0\), the equation has no real number solutions.

Answer:

The discriminant is equal to \(-16\), which means the equation has no real number solutions. (The first option: "The discriminant is equal to \(-16\), which means the equation has no real number solutions.")