Form 4 Chapter 2
Friday, February 21, 2014 | 12:09 AM | 0 Rain[s]
2.1 Quadratic expressions
* Identifying quadratic expressions
1. A quadratic expression in the form ax2+bx+c,where a,b and c are constants,a≠0 and × is an unknown.
For example:
(a) 3x2 - 4x + 5
(b) 2x2 + 6x
(c) x2-9
For example:
(a) 3x2 - 4x + 5
(b) 2x2 + 6x
(c) x2-9
In a quadratic equations :
- there is only one unknown
- the highest of the power unknown is 2 .
* Forming quadratic expressions by multiplying two linear expressions1. When two linear expressions with the same unknown are multiplied,the product is a quadratic expression.
2. The multiplication process is known as expansion* Forming quadratic expressions based on specific situationsTo form quadratic expressions based on specific situations :
1.Choose a letter to represent the unknown.2.Form a quadratic expressions based on the information given.
2. The multiplication process is known as expansion* Forming quadratic expressions based on specific situationsTo form quadratic expressions based on specific situations :
1.Choose a letter to represent the unknown.2.Form a quadratic expressions based on the information given.
2.2 Factorisation of quadratic expressions
*Factorising quadratic expressions of the form ax2 + bx + c,where b =0 or c =01.When b = 0, ax2 + c can be factorised by finding the highest common factor (HCF) of the coefficients a and c.
2.When c=0,ax2x< + bx can be factorised by finding the highest common factor of the coefficients a and b.x is also a common factor of the two terms.
*Factorising quadratic expressions of the form px2 - q,where p and q are perfect squaresLet p = a² and q = b²
px² - q = (ax)² - b²
= (ax + b) (ax - b)
*Factorising quadratic expressions of the form ax2 + bx + c, where a ≠ 0, b ≠ 0 and c ≠ 01.We can use the inspection method and cross method to factorise quadratic equations of this form.
2.To factorise quadratic expressions :
(a) ax² + bx + c, where a = 1
x² + bx + c =(x + p) (x + p)
=x² + qx + px + q² =x² + (p + q) x + pqIn comparison,
b = p + q ,c = pq
- Find the combination of two numbers whose product is c.
- Choose the number combination from step 1 whose sum is b.
(b) ax² + bx + c, where a > 1
ax² + bx + c =(mx + p) (nx + q)
=mnx² + mqx + npx + pq
=mnx² + (mq + np) x + pq
In comparison,
a = mn , b = mq + np , c = pq
- Find the combination of two numbers whose product is a.
- Find the combination of two numbers whose product is c.
- Choose the number combination from step 1 and step 2 whose sum is b.
*Factorising quadratic expressions containing coefficients with common factors For quadratic expressions containing coefficients with common factor first before
carrying out the factorisation of the expressions.
2.3 Quadratic Equations
* Indentifying quadratic equations with one unknown1. Quadratic equations with one unknown are equations involving quadratic expressions.
2. In a quadratic equation :
*Factorising quadratic expressions of the form ax2 + bx + c,where b =0 or c =01.When b = 0, ax2 + c can be factorised by finding the highest common factor (HCF) of the coefficients a and c.
2.When c=0,ax2x< + bx can be factorised by finding the highest common factor of the coefficients a and b.x is also a common factor of the two terms.
*Factorising quadratic expressions of the form px2 - q,where p and q are perfect squaresLet p = a² and q = b²
px² - q = (ax)² - b²
= (ax + b) (ax - b)
*Factorising quadratic expressions of the form ax2 + bx + c, where a ≠ 0, b ≠ 0 and c ≠ 01.We can use the inspection method and cross method to factorise quadratic equations of this form.
2.To factorise quadratic expressions :
(a) ax² + bx + c, where a = 1
x² + bx + c =(x + p) (x + p)
=x² + qx + px + q² =x² + (p + q) x + pqIn comparison,
b = p + q ,c = pq
- Find the combination of two numbers whose product is c.
- Choose the number combination from step 1 whose sum is b.
Quadratic Expressions and Equations
(b) ax² + bx + c, where a > 1
ax² + bx + c =(mx + p) (nx + q)
=mnx² + mqx + npx + pq
=mnx² + (mq + np) x + pq
In comparison,
a = mn , b = mq + np , c = pq
- Find the combination of two numbers whose product is a.
- Find the combination of two numbers whose product is c.
- Choose the number combination from step 1 and step 2 whose sum is b.
*Factorising quadratic expressions containing coefficients with common factors For quadratic expressions containing coefficients with common factor first before
carrying out the factorisation of the expressions.
2.3 Quadratic Equations
* Indentifying quadratic equations with one unknown1. Quadratic equations with one unknown are equations involving quadratic expressions.
2. In a quadratic equation :
- there is an equal sign "="
- there is only one unknown
- the highest power of the unknown is 2
2.4 Roots of Quadratic Equations
* Determining the roots of a specific quadratic equationThe roots of a quadratic equation are the values of the unknown which satisfy the quadratic equation.
* Determining the solutions for quadratic equations1.The solutions for a quadratic equation can be determined by :
* Determining the roots of a specific quadratic equationThe roots of a quadratic equation are the values of the unknown which satisfy the quadratic equation.
* Determining the solutions for quadratic equations1.The solutions for a quadratic equation can be determined by :
- trial and improvement method
- factorisation
2.To determine the solutions for ax2 + bx + c = 0 using trial and improvement method :
- try use the factors and the last term , c
