Mathematics Kenya
Quadratic Expressions and Equations
A group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation
Quadratic Expressions and Equations
Quadratic Expressions and Equations
Factorisation of quadratic expressions
Perfect squares
Completion of the square
Solution of quadratic equations by completing the square
Completing the square.
Example 1
Add a term that will make the following expression a perfect square: x2 - 8x
find a term that must be added to make the following expression a perfect square: x2 + 10x
Example 2
solve the following quadratic equation by completing the square: x2 + 4x + 1 = 0
Example 3
solve by completing the square: 3x2 + 7x – 6 = 0
Quadratic formula x=(-b±√(b^2-4ac))/2a
The special quadratic formula used for solving quadratic equation is:
Solution of quadratic equations using the formula
Example 4
solve 5x2 – 8x + 3 = 0 by using quadratic formula.
Example 5
solve this quadratic equation by using quadratic formula: 3x2 = - 7x - 4
Word problems leading to quadratic equations
Given a word problem; the following steps are to be used to recognize the type of equation.
Step1: choose the variables to represent the information
Step 2: formulate the equation according to the information given
Step 3: solve the equation by using any of the method you know
In order to be sure with your answers, check if the solution you obtained is correct.
Example 6
the length of a rectangular plot is 8 centimeters more than the width. If the area of a plot is 240cm2, find the dimensions of length and width.
Solution
Let the width be x
The length of a plot is 8 more than the width, so the length of a plot be x + 8
We are given the area of a plot = 240cm2 and the area of a rectangle is given by length ×width
then (x + 8) × x = 240
x2 + 8x = 240
rearrange the equation
x2 + 8x – 240 = 0
then solve the equation to find the value of x
Solving by splitting the middle term, two numbers whose product is -240 and their sum is 8, the number
are -12 and 20
our equation becomes; x2 + 20x – 12x – 240 = 0
x(x + 20) – 12(x + 20) = 0
either (x - 12) = 0 or (x + 20) = 0
x = 12 or x = -20
since we don’t have negative dimensions, then the width is 12cm and the length is 12 + 8 = 20cm
Therefore the rectangular plot has the length of 20cm and the width of 12cm.
Example 7
A piece of wire 40cm long is cut into two parts and each part is then bent into a square. If the sum of the areas of these squares is 68 square centimeters, find the lengths of the two pieces of wire.
Exercise 1
1. Solve each of the following quadratic equations by using factorization method:
- -6x2+ 23x – 20 = 0
- X2– x -12 = 0
2. Solve these equations by completing the square:
Formation of quadratic equations and solving them
Tables of values for a given quadratic relation
Graphs of quadratics equations
Simultaneous equations - one linear and one quadratic
Application to real life situation: Interpret the discriminant i.e. √(b^2-4ac)
Listening to this topic
