26th June 2003. What is the Function of the Quadratic?
The Author remembers his maths teacher enunciating the term Quadratic Function, What either word meant was wrapped in mystery. Now I know it comes from ‘Quadrate’, square or rectangle.
Why study quadratic equations?, was the subject in the House of Commons Today in 2003 when the Parliamentary Record, Hansard records a speech, by Mr McWalter MP, in a debate on science.
His theme was that a science dependent culture is not replenishing itself, and that a scientific basic is needed to continue its existence.
The speech was in response to a maths teacher, Mr. Bladen, of the National Association of Schoolmasters, quoted as saying that pupils be allowed to drop maths at the age of 14. He used Quadratic Equations as an example of the irrelevance of the subject for many.
Mr. McWalter, developed his argument by using these equations as an example of how tough areas of the curriculum should be tackled, or we end up ‘jumping over mole hills, instead of climbing mountains.’
Certainly one would be met by a blank face if confronted by a question as to the relevance of the Function, despite its practical applications going back to antiquity.
Many professions are said to be avid users, and in the Halls of Academia are important in computer modelling, in such varied explorations involving questions of product profitability, time, distance, trajectory and speed.
Basically Quadratic Equations, like all maths is a tool applicable for use in all human endeavour.
They are more complicated than linear and simultaneous equations, and have only one unknown, but allow it to be raised to a power of 2: as in the example, 3x² +x -10 =0.
As the saying goes, ‘there are many ways to skin a cat’ so, these equations can be solved by various means .(1)
(1) Apart from factoring, other methods include: completing the square, using a formula or finding the square root.
Solving by factoring example: x2-4x-12 =0. We factorize by using ‘FOIL’ Method (see below): (x-6)(x+2) =0. Then x2+2x-6x-12. Then x2-4x-12=0.
So since when we put x’s together the answer is 0, we know that either (x-6)=0+6 or (x+2)=0-2: so x=6 or x= -2 [minus 2].
In these examples there is always a positive and negative answer, so takes the answer which makes sense.
In equation x2+ 5x [middle term]+6 [constant]
We need to find the factors of 6 which add-up to 5. Since 6 can be written as a product of 2 and 3 and since 2+3=5 then I will use 2 and 3.
So (x+2)(x+3=x2+5x+6. Now we see how the factors =the equation.
FOIL METHOD for finding factors in quadratic equations, by finding products of each binomial (two sets of brackets), which could be for example (x-6)(x+2) =0.
F: product of first two numbers in the brackets.
O: product of outside two.
I: product of inside two.
L : product of last two.
(B) Solving by Completing the Square: x2-4x-12=0. X2-4x=0+12. X2-4x =12. X2-4x+4=12+4. Now by squaring 4x shared by 4 =4, we add 4 to both sides.
So now we get (x2-4x+4=square root of 16, therefore factors of this = x-2 = square root of 16. Then: x-2 =plus 4+2 or minus 4+2 so x =4+2 = 6 OR x =minus 4 +2 which equals minus 2. QED.
Ref: Purple Math, factorising quadratics the simple case.
Ref: Hansard Cols 1259-1267: 26.6.2003.
Ref: mathsisfun.com. quadratic-equations-real-world.
Ref: top image: duinkquadratics,wikispaces.com.
Ref: bottom image: googleimage.