Answer:
The quadratic equation
�
+
�
=
�
x+x=p can be simplified to
2
�
=
�
2x=p, or
�
=
�
2
x=
2
p
. Therefore, we are looking for values of
�
p such that
(
�
2
)
2
−
�
<
0
(
2
p
)
2
−p<0.
Simplifying this inequality, we get:
\begin{align*}
\left(\frac{p}{2}\right)^2 - p &< 0 \
\frac{p^2}{4} - p &< 0 \
p\left(\frac{p}{4}-1\right) &< 0 \
\end{align*}
To solve this inequality, we need to consider two cases:
Case 1:
�
4
−
1
>
0
4
p
−1>0, which means
�
>
4
p>4.
In this case, the inequality is satisfied when
�
p lies in the interval
(
4
,
∞
)
(4,∞).
Case 2:
�
4
−
1
<
0
4
p
−1<0, which means
�
<
4
p<4.
In this case, the inequality is satisfied when
�
p lies in the interval
(
0
,
4
)
(0,4).
Therefore, the values of
�
p that make the expression
�
+
�
=
�
x+x=p have no real roots are those that lie in the interval
(
0
,
4
)
∪
(
4
,
∞
)
(0,4)∪(4,∞), or equivalently,
�
∈
(
−
∞
,
0
)
∪
(
0
,
4
)
p∈(−∞,0)∪(0,4).