Cambridge Additional Mathematics

Quadratics (Chapter3) 65

QUADRATICS

Aquadratic equationis an equation of the form ax^2 +bx+c=0 wherea,b, andcare

constants, a 6 =0.

Aquadratic functionis a function of the form y=ax^2 +bx+c, a 6 =0.

Polynomial function Type

y=ax+b, a 6 =0 linear

y=ax^2 +bx+c, a 6 =0 quadratic

y=ax^3 +bx^2 +cx+d, a 6 =0 cubic

y=ax^4 +bx^3 +cx^2 +dx+e, a 6 =0 quartic

Quadratic functions are members of the family

ofpolynomials. The first few members of this

family are shown in the table.

Acme Leather Jacket Co. makes and sellsxleather jackets each

week. Their profit function is given by

P=¡ 12 : 5 x^2 + 550x¡ 2125 dollars.

How many jackets must be made and sold each week in order to

obtain a weekly profit of $ 3000?

Clearly we need to solve the equation:

¡ 12 : 5 x^2 + 550x¡2125 = 3000

We can rearrange the equation to give

12 : 5 x^2 ¡ 550 x+ 5125 = 0,

which is of the form ax^2 +bx+c=0 and is thus a quadratic equation.

SOLVING QUADRATIC EQUATIONS

To solve quadratic equations we have the following methods to choose from:

If ab=0then a=0or b=0.

² complete the square

² use thequadratic formula

² usetechnology.

Therootsorsolutionsof ax^2 +bx+c=0 are the values ofxwhich satisfy the equation, or make it

true.

For example: Consider x^2 ¡ 3 x+2=0.

When x=2, x^2 ¡ 3 x+ 2 = (2)^2 ¡3(2) + 2

=4¡6+2

=0 X

So, x=2is a root of the equation x^2 ¡ 3 x+2=0.

A Quadratic equations

² factorisethe quadratic and use the rule:

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