Let's start by understanding the roots of a quadratic equation and the given conditions. The roots of the quadratic equation are given by:
[tex]\[ x = \frac{-n \pm \sqrt{n^2 - 4mp}}{2m} \][/tex]
where [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( p \)[/tex] are positive real numbers. Additionally, [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( p \)[/tex] form a geometric sequence.
### Step 1: Expressing Terms in a Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. Let the common ratio be [tex]\( r \)[/tex].
Given [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( p \)[/tex] in geometric sequence, we have:
[tex]\[ n = m \cdot r \][/tex]
[tex]\[ p = n \cdot r = m \cdot r^2 \][/tex]
### Step 2: Substituting Geometric Sequence Terms
Substitute these expressions into the discriminant of the quadratic equation, [tex]\( n^2 - 4mp \)[/tex]:
[tex]\[ n = mr \][/tex]
[tex]\[ p = mr^2 \][/tex]
Now, substitute these into the discriminant:
[tex]\[ n^2 - 4mp = (mr)^2 - 4 \cdot m \cdot (mr^2) \][/tex]
### Step 3: Simplifying the Discriminant
Simplify the discriminant:
[tex]\[ (mr)^2 = m^2r^2 \][/tex]
[tex]\[ 4mp = 4m(mr^2) = 4m^2r^2 \][/tex]
Therefore, we have:
[tex]\[ n^2 - 4mp = m^2r^2 - 4m^2r^2 \][/tex]
[tex]\[ n^2 - 4mp = m^2r^2(1 - 4) \][/tex]
[tex]\[ n^2 - 4mp = -3m^2r^2 \][/tex]
### Step 4: Determining the Nature of Roots
The discriminant of a quadratic equation determines the nature of its roots. The discriminant here is:
[tex]\[ -3m^2r^2 \][/tex]
Since [tex]\( m \)[/tex] and [tex]\( r \)[/tex] are positive real numbers, [tex]\( m^2r^2 \)[/tex] is a positive real number. Multiplying by [tex]\(-3\)[/tex] makes it negative:
[tex]\[ -3m^2r^2 < 0 \][/tex]
A negative discriminant implies that the quadratic equation has non-real (complex) roots.
### Conclusion
Therefore, given that [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( p \)[/tex] form a geometric sequence, the discriminant of the quadratic equation [tex]\( n^2 - 4mp \)[/tex] is negative, leading to non-real number roots for [tex]\( x \)[/tex].
Thus, we have proven that [tex]\( x \)[/tex] is a non-real number.
