Algebra
Clement Radcliffe, Contributor

THIS WEEK we will complete the review of Algebra by considering aspects of graphs. Specifically, it is my intention to elaborate on the solution of quadratic equations using a graph.

REMINDERS

* A quadratic equation is represented graphically by a curve

* A curve should be drawn by free hand sketch

* The x axis has the equation y = 0 and the y axis has equation x = 0.

* Given the curve y = f(x) and the line y = g(x), then the points of intersection of both are represented by:

y = f(x) = g(x) ... f(x) = g(x)

If f(x) = x² - 3x + 2 and g(x) = 2x - 1

Then at the point of intersection of the curve and the line f(x) = g(x).

... x² - 3x + 2 = 2x -1

... x² - 5x + 3 = 0

The x coordinates of the points of intersection will represent the solution of the equation x² - 5x + 3 = 0.

EXAMPLE

Using an appropriate scale please plot the curve y = 3x² - 2x - 1. Hence solve the equations:

a) 3x² - 2x -1 = 0

b) 3x² - 1 = 2 - 2x

c) 3x²- 3 = 0 or x² - 1 = 0

a) Given the curve y = 3x² - 2x - 1, the solution of the equation

3x² - 2x -1 = 0 is the x values of the points of intersection of the curve

y = 3x² - 2x -1 and the line

y = 0 or the x axis.

... The solution is x = 1, -33

b) Given the curve y = 3x² - 2x -1 by plotting the line y = 2 - 2x, then the points of intersection of the curve and the line will represent the solution of the equation

3x² - 2 y -1 =2 - 2x.

From the graph the solution is

x = -1, 1.

c) Given the equation

3x² - 3 = 0, if the curve

y = 3x² - 2x -1 must be used, then

3x² - 3 = 0 is reorganised as follows:

3x² - 2x + 2x - 2 -1 = 0

... 3x² - 2x -1 =2 - 2x

... The solution of the equation

3x² -3 = 0 is the x coordinates of the points of intersection of the curve y = 3x² - 2x -1 and the line

y = 2 - 2x.

... As in b) the solution is x = -1, 1.

Let us attempt another example.

Given the curve

y = 2x² - x -3, solve the equation 2x² - 2x - 5 = 0.

By reorganising the equation

2x² - 2x - 5 = 0, it follows that:

2x² - x - x -3 - 2 = 0

... 2x² - x - 3 = x + 2

Then the solution of

2x² - 2x -5 = 0 is the coordinates of the points of intersection of the curve

y = 2x² - x - 3 and the line

y - x +2.

Please continue to solve similar examples from the past papers. This section two question must be done if it appears on your exam paper in June.

* Clement Radcliffe is principal of Glenmuir High School in Clarendon.