Expert Mathmatician
Quadratic Examples
Max/Min problems:
For example you can have a problem such as: A baseball's fligh path is modelled by the equation t=5m^2+4m+2 where t represents the time and m represent the distance in metres.
If the problem asks you to find the max height, then what you need to do is substitute these points into Vertex form. To do this you need to Complete the Square.
Finally, If the problem asks you to find the point at which the ball landed you will need to sub the points into the quadratic formula OR sub y=0 and solve for x
If the problem asks you to find the initial height the ball was at, you can uses the c term (in this case:2) to get the initial height. This is because you get the initial height when x=0.
Black words indicate the steps
Orange words indicate the example
Bolded words indicate concepts or key information
Area/Length Problems:
Lets say we had a rectangle with an area of 20m^2. The width is 3m more than the length. find the two dimensions.
First, you would need to write your let statements. (let l represent the length and let w represent the width).
Then, write down the two equations needed for the problem ( 1.w=l+3 & 2.l*w=20).
Now, substitute one of the equations into the other. lets use the 1st one into the 2nd one.(l*(l+3)=20).
Now, bring the answer to the other side. After collecting like terms you will have a standard form equation (l^2+3l-20=0).
Finally, the Quadratic formula and solve for the zeroes or you can substitute y=0 if one of the numbers is a positive and the other is a negative, the positive one is correct as there can not be negative length.
Length problems for Triangles:
If we had a problem such as:
A triangle has a hypotenuse of the length 25cm. The smallest side is 3cm less than the second side
First let us define our variables (let s represent the small side. Let m represent the second side).
Now, let's make the equations (recall from grade 9 the Pythogoream Theorm (1. m=s+3 & 2. s^2+m^2=c^2).
Next, substitute one of the equations into the other. we will use the 1st one into the 2nd one.(s^2+(s+3)^2=25^2).
Now, bring the answer to the other side and collect like terms after you have simplified it (e.g square the numbers etc.).
Finally, by having an equation in Standard Form you will need to find the x-intercepts. Use the Quadratic Formula or substitute y=0 to find the x-intercepts. As i stated above, there can not be negative lengths!
Consecutive numbers:
For this lets have a problem like, the product of two consecutive numbers is equal to 20.
For starters, consecutive numbers are numbers that are right after each other like 1,2,3 etc.
First lets define our variables (1. let x represent the smaller number & 2. let y represent the larger number).
Next lets define our two equations (1. y=x+1 & 2. x*y=43)
Now we will substitute one of the equatiuons into the other. I like to do the 1st one into the second (x(x+1)=20).
Then after simplifing the equation and bringing the answer to the other side, we will have an equation in Standard Form.
Finally, we will need to find the X-intercepts using the Quadratic Formula or Subbing y=0.
Note:this time the answer can be a negative or positive so after you get your x-intercept(s) you will take each one and sub it into your first equation (y=x+1)
Revenue:
Revenue questions are easy to do but take some thinking. lets take the question: Tickets to a school dance cost $2, and the projected attendance is 100 people. For every $1 increase in the ticket price, the dance committee projects that attendance will decrease by 10. what ticket price will generate $1200 in revenue?
The first step we will do is get our key information (the price is 2 and 100 people will come. for every 1$ increase 10 people wont come).
Let's make our variable now (let x represent the number of price decrease).
Now lets make our equations
(1. price= 2+1x & 2. revenue=100-10x).
To get the total revenue we will need to multiply price times revenue ((2+1x)(100-10x)).
Now use FO.I.L and you will have a Standard Form equation.
Finally use the Quadratic Formula or substitute y=0 to get the x-intercept.
Remember there is NO negative money!