7) Write the equation of the line tangent to the parabola with the equation y = x^2-3x + 5 and parallel to the line with the equation y = 2x. Then determine the coordinates of the intersection point. Do not use derivatives Solve by imposing zero discriminant and find k. To find the coordinates of the contact point solve the system formed by the parabola and the tangent line and find x0

Given parabola equation $y = x^2 - 3x + 5$ and line equation $y = 2x$.

Since the tangent line is parallel to $y = 2x$, the slope of the tangent line is also 2. Therefore, the parabola must have a point where the derivative of the parabola equals 2. We will solve this without using derivatives.

Let the point of tangency be $(a, b)$ where the tangent line intersects the parabola.

1. Equation of the tangent line at point $(a, b)$:
By point-slope form of the equation of a line: $y - b = 2(x - a)$

2. Substituting the given point of tangency $(a, b)$ into the equation of the parabola:
$b = a^2 - 3a + 5$

3. Substituting these results into the equation of the tangent line:
$y = 2x + 2a - (a^2 - 3a + 5)$
$y = 2x + 2a - a^2 + 3a - 5$
$y = 2x + (5 - a^2 + a)$

Since the tangent line is parallel to $y = 2x$, the respective coefficients of $x$ should be equal:
$2 = 2$
And the coefficients of $y$ and the constants should be equal:
$0 = 5 - a^2 + a$

4. Solve for $a$:
$5 - a^2 + a = 0$
$a^2 - a + 5 = 0$

The discriminant must be zero since the line is tangent to the parabola:
$\Delta = (-1)^2 - 4(1)(5) = 1 - 20 = -19$

Since the discriminant is negative, there is no real $a$ such that the tangent line is parallel to $y = 2x$ and tangent to the parabola.

Therefore, the equation of the line tangent to the parabola with equation $y = x^2 - 3x + 5$ and parallel to the line $y = 2x$ does not exist. The intersection point coordinates are not determinable.

\textbf{Answer:} The equation of the tangent line does not exist.