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Question 104690: use factoring to solve problem
a triangular sail has an area of x^2 + 5x + 6 square meters and a height of x + 3 meters. find the lenght of the sails base
Answer by TP(29)   (Show Source):
You can put this solution on YOUR website! To do this you need to remember that
Area of triangle=(base*height)/2 and you should be able to factorise a quadratic trinomial expression such as ax^2+bx+c where a,b and c are given numbers(constants).You will need to be able to factorise x^2+5x+6.
Let the base be b metres(this is how we spell this word in "English" English).
Then Area of Triangle = [b*height]/2
Since the height is given as (x+3) metres then
Area=[b*(x+3)]/2.
But you are told that the area is x^2+5x+6
So [b*(x+3)]/2 is the same as x^2+5x+6
And so we can write
[b*(x+3)]/2=x^2+5x+6
Now the Left Hand Side of the equal sign is in a factorised form ,that is it is written as one thing times another thing (b times (x+3))
So to be able to find out what b is we need to factorise the Right Hand Side which is x^2+5x+6.
Suppose the factors are x+a and x+b then (x+a)*(x+b) =x^2 +(a+b)x+ab [Multiplying the brackets together]
Now compare x^2+(a+b)x+ab (i) to x^2+5x+6 (ii).
You can see that a+b in expression (i) is in the same place as the number 5 is in expression (ii)
and where the ab is in (i) we have the number 6 in (ii)
And so we can write
a+b=5 and
a*b=6 [this is called comparing coefficients]
So to be able to write x^2+5x+6 as (x+a)*(x+b) we need to find two numbers (a and b) so that when you add them together you get 5 and when you multiply them together you get 6.
The numbers are then a=3 and b=2 (3+2=5 and 3*2=6)
This means then that we can write the equation [b*(x+3)]/2=x^2+5x+6 as
[b*(x+3)]/2=(x+3)*(x+2)
Now share both sides by(x+3) and we get
b/2=(x+2)
Now multiply both sides by 2 and we get
b=2*(x+2)
Now multiply the bracket out(multiply everything in the bracket by 2) and we get
b=2x+4
And so the base is 2x+4 metres ANS
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