SOLUTION: a) Solve the simultaneous equation : 𝑦 = 𝑥^2 − 3𝑥+2 and 𝑦 = 3𝑥−7 b) Interpret your solution to part (i) geometrically

y = x^2 − 3x + 2  represents a parabola.

y = 3x - 7        represents a straight line.


Simultaneous solution represents the intersection points.


From  y = x^2 − 3x + 2  and  y = 3x - 7  we get, by substitution

    x^2 - 3x + 2 = 3x - 7.


Collecting the terms on the left side, we transform it to the standard form quadratic equation

    x^2 - 6x + 9 = 0.


Left side is a full square

    (x-3)^2 = 0.


So, x= 3 is the solution to the quadratic equation of multiplicity 2.

In other words, two solutions of the quadratic equation do merge in one single point.

Geometrically, it means  that the straight line  y = 3x-7  is a tangent to the parabola.


The tangent point is x= 3,  y= 3x-7 = 3*3-7 = 2.