1. Solving by Factoring
2D Math

2. Equation types The STANDARD FORM of a quadratic function is
The FACTORED FORM of a quadratic function is
To change from the standard form to the factored form, factor the quadratic.
To change from the factored form to the standard form, expand and simplify.

3. Example as a note!
To raise money for Student Council, they sell T-shirts.
Based on previous years’ sales, they can sell 40 T-shirts a week at $10.
If they raise the price by $1, they will sell one less T-shirt a week.
They picked some prices, estimated the number of shirts they might sell and calculated the revenue.
Revenue is the Price per shirt x # of T-shirts sold

4. Chart Price ($) T-shirts Sold Revenue ($) 10 40 400 10 + 10 = 20
= 25
= 40

5. Chart Price ($) T-shirts Sold Revenue ($) 10 40 400 10 + 10 = 20
40 – 10 = 30
30 x 20 = 600
= 25
40 – 15 = 25
25 x 25 = 625
= 40
40 – 30 = 10
40 x 10 = 400
You will notice that revenue increased and then decreased.
A quadratic function can model the revenue.

6. Questions
1) If x represents the number of $1 increases and R(x) is the revenue:
a) Write an expression for the # of t-shirts sold
# of T-shirts = (40 – x)
b) Write an expression for the price per t-shirt
Price per shirt = (10 + x)
c) Write a function for the revenue
R(x) = (40 – x) (10 + x)
This is – (x – 40) (x + 10) in factored form (a = -1).

7. Questions
2) Use the factored form to determine the zeros of the function.
The zeros (or x-intercepts) of a function are the values of x for which the function has the value zero.
To find the x-intercepts (if no graph is plotted):
a) Set the factored form of the function equal to 0:
– (x – 40) (x + 10) = 0
b) Set each factor equal to 0:
(x – 40) = 0 and (x + 10) = 0
c) Solve for x in each factor:
x = 40 and x = - 10
These are the zeroes (roots) or x-intercepts.

8. Questions 3) Determine the maximum revenue.
To find the maximum (or minimum value), determine the vertex.
The maximum (or minimum value) is the y-coordinate of the vertex and the vertex lies on the axis of symmetry.
To find the axis of symmetry (which is also the x-coordinate of the vertex), add the zeros of the function together and divide by 2.
Axis of symmetry: {40 + (- 10)}/2 = 15 Equation: x = 15

9. Question 3 contd
As a = -1, the quadratic opens down, thus the vertex is at a maximum.
The vertex occurs at x = 15, so the maximum revenue is generated when x is 15.
R(15) = - (15)2 + 30(15) + 400 =
= 625
This agrees with the chart from earlier, and the graph we plotted.