Answer:
To find the range of values of k for which the equation \(3x^2 - 7x + k = 0\) has no real roots we can examine the discriminant of the quadratic equation.
The discriminant (\(\Delta\)) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\).
In our given equation \(3x^2 - 7x + k = 0\ the coefficients are: a = 3 b = -7 and c = k.
Using the quadratic formula the solutions to the equation are given by:
\[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]
For the equation to have no real roots the discriminant must be negative (\(\Delta < 0\)). So we have:
\[b^2 - 4ac < 0\]
\[(-7)^2 - 4 \cdot 3 \cdot k < 0\]
\[49 - 12k < 0\]
\[12k > 49\]
\[k > \frac{49}{12}\]
Therefore the range of values for k that makes the equation \(3x^2 - 7x + k = 0\) have no real roots is \(k > \frac{49}{12}\).