Parabolas and Intersecting Lines

Suppose you have already plotted P₁: y = x² - 3x + 1, but the teacher asks you to find graphically the roots of P₂: y = x² - 4x + 3. What a bother – now you need to plot another parabola. Or do you?

Johnny can rearrange P₂ to get an equation of the form: sloped line = parabola already plotted.

0 = x2 - 4x + 3	Given P2
+x =( x2 - 4x + 3) +x	Add 'x' to both sides
x = x2 - 3x + 3	Simplify by collecting ‘x’ terms on the right
x - 2 = (x2 - 4x + 3) - 2	Add '-2' to both sides, i.e., subtract
x - 2 = x2 - 3x + 1	Simplify by collecting constant terms on the right
line = given P1

Now Johnny has a sloped line = parabola that he has already graphed

The sloped line has m = 1 and c = -2, so it is easy to draw.

Reading from the graph, the solution to the teacher’s question is x = 1 or x = 3.

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As a check, Mary factorises y= x² - 4x + 3

y = (x – 3) (x – 1)

When y = 0,

x – 3 = 0, so x = 3

or x – 1 = 0 so x = 1

Done!