quadratics lecture notes

Topic 1: Quadratics

Curriculum Content:

1.1 General form of a quadratic equation

A quadratic equation with one variable, x is an equation containing x2 as the highest power

of x.For instance , x2 – 4x = 1 , 2x2 + 4 = 0 and 3x2 – 5 = 0.

A quadratic equation expressed in general form is written as:

The value of x which satisfies the quadratic equation is known as “__________”

A quadratic equation has “_______” roots at most.

1.2 Solving quadratic equations

Solving quadratic equation ax2 + bx + c = 0 means finding the roots of the equation.

The roots of a quadratic equation can be solved by :

a) Factorisation

b) completing the square

c) using the formulae :

a) Factorisation

Solve the quadratic equation by factorisation:

x2 + 2x -15 =0

2x2 = 13x – 6

b) Completing the square:

Take a quadratic equation ax2 + bx + c = 0 and convert it to a( x + p ) 2 + q = 0 where a,

p, and q are constants using the method of completing the square.

Example 1:

Write the following quadratic equation in a( x + p ) 2 + q = 0 form

x2 + 8x – 2 = 0

Example 2:

Express x2 = 6 – 2x in the form a( x + p ) 2 + q = 0

Hence solve x2 = 6 – 2x

c ) Quadratic formulae

By simply using the quadratic formulae

a quadratic equation can be solved.

Exercise

1.3 Determining the nature of the roots from the value of b2 – 4 ac

The nature of the roots of a quadratic equation ax2 + bx + c = 0 is determined by the

expression b2 – 4 ac which called the discriminant of the equation. Three possibilities can

arise:

a) b2 – 4 ac > 0 (positive)

The equation has two distinct or different root.(real roots)

b) b2 – 4 ac = 0

The equation has two equal roots.

c) b2 – 4 ac < 0 (negative)

The equation has no real roots.

Exercise 1

a) If the roots of the equation 2x2 – px + 8 = 0 are equal, find the value of p.

b) Prove that the equation (k-2)x2 + 2x – k = 0 has real roots whatever the value of k.

c) Prove that the roots of the equation kx2 + ( 2k + 4 )x + 8 = 0 are real for all values of

k.

d) Show that the equation ax2 + ( a + b )x + b =0 has real roots for all values of a and b.

e) Find the relationship between p and q if the roots of the equation px2 + qx + 1 = 0 are

equal.

1.4 Shape of Graphs of Quadratic Functions

The shape of the graph of a quadratic function f(x) = ax2 + bx + c is a smooth symmetrical

curve. The curve is known as a parabola.

The position of the graph of f(x) = ax2 + bx + c depends on the type of the roots of f(x) = 0. If

the position of the graph of f(x) = ax2 + bx + c is given, then the type of roots for f(x) = 0 can

be determined.

Example:Sketch these simple quadratic functions on the same axes.

i) f(x) = x2

ii) f(x) = 2x2

iii) f(x) = ½x2

iv) f(x) = -x2

1.5 Sketching Graphs of Quadratic functions

To draw the graph f(x) = ax2 + bx + c, we need a table of values. For a sketch, we only need

to know:

The shape of the graph, by determining the value of a

The position of the graph by determining the value of the term b2 – 4ac

The maximum or minimum point of f(x) = ax2 + bx + c, by expressing f(x) in terms of

a ( x + p )2 + q. ( -p , q ) is a maximum or minimum point and the graph is symmetrical

about the axis through the point ( -p , q ).

The position of the roots ( if any ) of f(x) = 0

Where the graph intersects the y-axis. This is given by f(0).

Mark all the points found on a Cartesian plane and draw a smooth parabola passing through

all the points.

Example 1: Given that the quadratic function f(x)= 5x – 2 – 3x2,Express f(x) in the forms of

m(x + n )2 + p, where m,n, and p are constants. Determine whether f(x) has a maximum or

minimum point and state its value. Hence sketch the graph of f(x).

Example 2: Given y=h + 2kx-x2 = p-(x + q)2.

a) Find the value of

(i) p

(ii) q

in terms of h and/or k.

b) If k=3, state the axis of symmetry of the curve.

c) It is given that the line y=4 touches the curve y= h + 2kx –x2.

(i) State h in terms of k

(ii) Hence, sketch the graph of the curve.

Example 3: Solve 3x2-7x + 4 = 0.Hence, sketch the graph of the quadratic equation and state

the axis of symmetry.

Example 4: Sketch the graph of I x2 – 4x + 3 I and find the range of values of y for 0≤ x ≤2.

Example 5: Sketch the graph of I x (1-x) I and find the range of values of f(x) for -1≤ x ≤1/2.

1.6 Solving Simultaneous equations

Exercise:

1. Solve the following pairs of simultaneous equations.

a) y= x + 1, x2 + y2 = 25

b) x + y = 7 , x2 + y2 = 25

c) y=x -3 , y=x2-3x-8

d) y=2-x , x2-y2 = 8

e) 2x + y = 5, x2+ y2 = 25

f) y= 1 –x, y2 - xy= 0

Example 1: Solve simultaneous equation y=x2, x + y = 6

2.Find the coordinates of the points of intersection of the given straight line with the given

curves.

a) y= 2x + 1, y=x2-x +3

b) y= 3x + 2 , x2 + y2 = 26

c) y= 2x -2 , y=x2 – 5

d) x + 2y = 3, x2 + xy = 2

e) 3y + 4x = 25, x2 + y2 = 25

f) y + 2x = 3, 2x2 - 3xy = 14

3.In each case find the number of points of intersection of the straight line with the curve.

a) y= 1 – 2x, x2 + y2 = 1

b) y= ½ x – 1 , y= 4x2

c) y= 3x -1 , xy = 12

d) 4y – x =16, y2 = 4x

e) 3y-x=15, 4x2 + 9 y2 = 36

1.7 Equations which reduce to quadratic equations

Sometimes you will come across equations which are not quadratic, but which can be

changed into quadratic equations, usually by making the right substitution.

Example: Solve the equation x4- 13 x2 +36 =0

Example : Solve the equation √x = 6-x

Exercise:

Question 6

Quadratic Inequalities questions