√[a ² + b² + 2√(ab)]=?
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A quadratic function has an x² term, its general form is:
a x² + b x + c = 0
A quadratic equation is one in which the unknown is in the second degree. Usually a quadratic has two solutions or roots:
eg if:
x² = 36
x = √36
x = +6 or -6
can be written as:
x = ± 6
Methods of solution
by factors
by completing the square
by formulae
by graph
Factorising
The first step in factorising is to find the highest common factor in expressional terms
4 l² x² - 2 l x3
2 l x² (2 l - x)
Factors of a² - b²
a² - b² = (a+b)(a-b)
This can be used to factorise the difference of any two squares
Factorising
solve the equation
3 x² = 2 x +5
3 x² - 2 x - 5 = 0
(3x - 5)(x+1) =0
so either (3x - 5) =0 or (x+1) =0
so either x = 3/5 or x=-1
completing the square
The basis of this method is the forming of a perfect square, examples of perfect squares are:
(x+1)² = x² + 2 x + 1
(x-4)² = x² - 8 x + 16
Consider the equation x² + 10x
To form a perfect square a number is added which is the square of 1/2 the coefiant of x i.e.:
x² + 10x + (10/2)²
= x² + 10x +25
= (x+5) ²
Another example
6 x² + 11 x = 10
first step: divide both sides by the coeficient of x²(ie 6)
6 x² /6 + 11/6 x = 10/6
second step: make a perfect square by adding to both sides the square of 1/2 the coeficient of x ie (11/12) ²
x² + 11/6 x + (11/12) ² = 10/6 + (11/12) ²
(x + 11/12) ² = 136/144
(x + 11/12) =± √(136/144)
x = - 11/12 ± 19/12
x = -30/12 or 8/12
x = -2.5 or 0.667
Solution by formula
if
a x² + b x + c = 0
then x= -b ± √(b² - 4ac )
2a
This formula can be proved by following the completing the square method with algebraic constants a,b and c instead of actual numbers:
a x² + b x + c = 0
a x² + b x = -c
x² + b/a x = -c/a
x² + b/a x + (b/2a)² = -c/a + (b/2a)²
(x + b/2a)² = -c/a + b²/4a²
x + b/2a = ± √(-c/a + b²/4a²)
quadratic formula