Solving quadratic equations
Clement Radcliffe, Contributor

If you have been following the materials presented in the last three lessons, you should realise, by now, that the following methods are commonly used to solve quadratic equations:

• Quadratic factors
• Quadratic formula
• Completion of square.

Learning each method is important. It is also critical that you know when to use each. Let us review the materials presented previously with this in mind.

• Only some quadratic equations can be solved by the factorisation method.
• Given the equation, you should first use the factorisation method, unless otherwise directed.
• If a specific method is requested, you must obey the instructions or you will be penalised.
• All quadratic equations with real roots (equations with real numbers as their solutions) can be solved using both the formula and completion of square methods.
• Be sure to use the correct formula and be careful in processing the negative signs when using the formula method.
• If you are asked to solve a quadratic equation correct to two decimal places, you should use the formula method.

Please continue to practise solving quadratic equations by attempting the following:

1. Solve the equation: a² - 8a +16 = 0

2. Solve the quadratic equation: 3x² - 5x - 4 = 0, giving your answer correct to two decimal places.

3. Express the equation: x² - 10x + 21 = 0, in the form. (x-a)² = b where a and b are constants.

4. Solve: 4x² + 3 = 8x, using the completion of square method.

Let us now turn our attention to the homework from the previous lesson.

Example 1

Solve the simultaneous equations:

y - x = 1
y = x² - 3x + 4

Solution

Given y - x = 1 ... (1)

y = x² - 3x + 4 ... (2)

From equation (1), y = x + 1

Substituting into equation (2),

... x + 1 = x² -3x + 4

... x² - 3x - x + 4 - 1 = 0.

... x² - 4x + 3 = 0 Factorising:

(x-1)(x-3) = 0

... x-1 = 0; x = 1

OR x-3 = 0; x = 3

Substituting in equation (1),

when x = 1, y = 2

when x = 3, y = 4

Answers: x = 1, y = 2 and x = 3, y = 4.

Example 2

Solve the simultaneous equations: x² + 9y² = 37

x - 2y = -3

Solution

Given x² + 9y² = 37 ... (1)

x - 2y = -3 ... (2)

Using equation (2)

x = 2y - 3 ... (3) Substitute in equation (1)

... (2y -3)² + 9y² = 37

... 4y² -12y +9 + 9y² = 37

13y² - 12y - 28 = 0 Solve using factorisation method

(13y + 14)(y - 2) = 0

... 13 + 14 = 0, ... y = -14/13

OR y - 2 = 0, ... y = 2.

Substituting values of y into equation (3)

... When y = - 14/13, x = (2 x -14 -3)/13

... x = (-28-3)/13 = -67/13

When y = 2, x = 2 x 2 -3 = 1.

Answer: x =1, y = 2 OR x = -67/13, y = -14/13

We will now complete algebra by reviewing aspects of graphs.

GRAPHS

Please be reminded that you are required to be able to draw straight-line and quadratic graphs. In doing so, it is important that you pay attention to the following:

• You need to complete accurately an appropriate table of x and y values.
• The x and y axes must be clearly labelled.
• The scale used must be appropriate to the problem. If one is given, it must be accurately used.
• A ruler must be used to draw the straight line while free hand must be used to draw the curve.
• The use of a suitable pencil (HB) is required.

APPLICATIONS

Graphs may be used to solve:

• Quadratic equations
• Simultaneous equations

In both cases, the solution is represented by the x and y coordinates at the points of intersection of the line and the curve.

EXAMPLE

Plot the equations y = 3x² - 2x - 1 and y = x + 5.

Hence: (a) Solve the equation 3x² - 2x - 1 = 0.

(b) Solve both equations simultaneously

Completing the tables:

y = 3x² - 2x -1

y = x + 5

(a) The solution of 3x² - 2x - 1 = 0 is the x coordinates of the points of intersection of the curve and the x axis.

As the x axis is y = 0, then at the points of intersection of y = 0 and y = 3x² -2x -1,

y = 3x² - 2x -1= 0. Therefore, the x values are: 1 and -.33

The solution of the equation 3x² -2x -1 = 0 is therefore, x = 1 OR - 0.33

Answer: 1 and -.33

(b) The points of intersection of the curve y = 3x² -2x -1 and the line y = x + 5 represent the solution of the simultaneous equations. Therefore, the solutions are x = -1, y = 4 and x = 2 , y = 7.

NB: At the points of intersection, 3x² - 2x -1 = x + 5.

Simplifying: 3x² -2x - x -1 -5 = 0

... 3x² -3x -6 = 0 or x² - x - 2 = 0

... the values of x above also represent the solutions of the equation x² - x - 2 = 0 .

We will continue the review of graphs next week.

Footballer Careme Gibson (right) is presented with a bag of goodies from Flow's PR Manager, Denise Williams, as his coach, Neville Bell, looks on during a special presentation to St George's College under Flow's building leaders through technology project at the school, recently. Flow donated free Internet access and a computer to the school. - Peta-Gaye Clachar/Staff Photographer

Clement Radcliffe is the principal of Glenmuir High School in May Pen.