To determine the value of [tex]\( n \)[/tex] such that the quadratic equation [tex]\( 2x^2 - 21x + c = 0 \)[/tex] has no real solutions if [tex]\( c > n \)[/tex], we start by examining the condition under which a quadratic equation has no real solutions.
Consider a general quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. The quadratic equation will have no real solutions if its discriminant is less than zero. The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the equation [tex]\( 2x^2 - 21x + c = 0 \)[/tex], we identify the coefficients as [tex]\( a = 2 \)[/tex], [tex]\( b = -21 \)[/tex], and [tex]\( c = c \)[/tex].
The discriminant for our equation is:
[tex]\[ \Delta = (-21)^2 - 4 \cdot 2 \cdot c \][/tex]
Simplifying the expression, we get:
[tex]\[ \Delta = 441 - 8c \][/tex]
For the quadratic equation to have no real solutions, the discriminant must be less than zero:
[tex]\[ 441 - 8c < 0 \][/tex]
Solving this inequality for [tex]\( c \)[/tex], we add [tex]\( 8c \)[/tex] to both sides:
[tex]\[ 441 < 8c \][/tex]
Then, we divide both sides by 8:
[tex]\[ \frac{441}{8} < c \][/tex]
Thus, the least possible value for [tex]\( n \)[/tex] is:
[tex]\[ n = \frac{441}{8} \][/tex]
Therefore, the quadratic equation [tex]\( 2x^2 - 21x + c = 0 \)[/tex] has no real solutions if [tex]\( c > \frac{441}{8} \)[/tex]. The least possible value of [tex]\( n \)[/tex] is:
[tex]\[ n = \frac{441}{8} \][/tex]
