1 Linear and Quadratic Equations An equation states that two expressions are equal. In solving an equation, operations are performed to isolate the variable or unknown. This produces equivalent equations.

2 Allowable operations include: adding or subtracting the same quantity from each side of the equation multiplying or dividing both sides of the equation by the same constant or some expression that includes the variable simplifying or expanding either side of the equation to assist in its solution raising both sides of an equation to the same power

3 Example 1a

4 Example 1b

5 Example 1c

6 Example 2

7 Linear Equations Linear equations can be written in the form Conditions: Feature: the power of the variable, in this case, x, is always exactly 1.

8 Often the equation will not be in the form but can be manipulated and expressed in this way. A good generalised model for solving equations is based on the process of  expansion  i solation  c onsolidation

9 Isolation Isolation is the process of rearranging an equation in such a way that the unknown is left by itself on one side of an equation. Take all the terms involving the unknown (just the 2x term) to the left and every other term to the right of the equals sign. We can do this by subtracting 9 from both sides:

10 Divide both sides by 2 to get the x by itself: Check that this is correct by substituting into the original equation:

11 Consolidation If the unknown appears more than once in an equation, consolidation is needed before proceeding to isolation. Take all the terms involving the unknown to the left by subtracting 2t from both sides:

12 Now use isolation as before:

13 Expansion When the unknown appears in a bracket, expansion is needed. Expand both sides: and simplify

14 Now consolidate: and isolate:

15 How many solutions?  always true – true for all values of the unknown(s)  sometimes true and sometimes false – true for some values of the unknown(s)  never true – true for no value of the unknown(s) An equation can be:

16 Example Now attempting to consolidate: which is true for all values of x.

17 Example Now attempting to consolidate: Since this is clearly not true, the equation has no solution.

18 Further examples are given in the study guide. Pay particular attention to Example 3-10 which shows what to do with messy or ugly equations.

19 Equations Leading to Linear Equations Consider the following equation In this form, it does not appear to be linear at all but with a number of simple operations it can be expressed as follows and hence solved.

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21 Section 0.7 of the text also contains excellent examples of equations leading to linear equations.

22 Of particular interest and providing the most challenge is the creation of the mathematical equation from some verbal description of a process.

23 A tank contains 25 litres of water but has the capacity for 1000 litres. Water begins pouring in at a rate of 15 litres per minute. How long will it take to fill the tank? Number of minutes = x Example 4

24 Consider a factory that produces moulded toys. It costs $3000 to set up a new mould and $5 for the materials to produce a single toy. If the toys can be sold for $10 each, how many need to be produced to break even? Linear Models Cost/revenue model of manufacture

25 The cost of manufacture can be represented by the expression 5x where x is the number of toys manufactured. This information is sometimes represented as the cost function and written as C(x) represents the total cost of producing x toys.

26 In general, C(x) = Bx+A is called the linear cost function. A is the total fixed costs, that is, the sum of all costs not dependent upon the number of units manufactured (cost of machines, rental of floor space, insurance etc). B is the total variable or per unit costs (materials used per unit, electricity cost per unit etc).

27 In a similar way, the revenue generated by toy sales is given by 10x and incorporated in the revenue function as R(x)=10x. In general, R(x)=Dx is called the linear revenue function where D is the unit selling price.

28 By definition, the break-even point is the point where costs equal revenue, that is, C(x)=R(x).

29 A simple check shows that when x = 600, C(x) = R(x) = $6000. It is easy to show that when x is less than 600, C(x) is greater than R(x), a loss... and that when x is greater than 600, C(x) is less than R(x), a profit.

30 Graphically, this can be shown as follows: 600 cost revenue $ x

31 The cost and revenue models are linear (can be graphed as straight lines), but in practice this may not be so. To increase production past a certain point may mean working overtime so per unit costs are no longer constant. Discounts for bulk orders may be given, etc. These situations require more elaborate models.

32 Quadratic Equations A quadratic equation can be written in the form Feature: the highest power of the variable is 2. Quadratic equations can be solved using  factoring  the quadratic formula Conditions:

33 Factoring Example 1

34 Example 2

35 The Quadratic Formula The solution to a quadratic equation is given by Method: 1. Manipulate the equation into the standard form. 2. Identify the value of the constants a, b and c. 3. Substitute them into the formula and calculate. There may be 0, 1 or 2 solutions. Why?

36 2 solutions 1 solution No solution

37 Algebraically, the situation is described as follows: is called the discriminant. When the discriminant is less than 0, there are no solutions to the equation since the square root of a negative number is not real. When the discriminant equals 0, there is one solution and when the discriminant is greater than 0, two solutions can be found.

38 Example 1 Quadratic Formula

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40 Example 2

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42 Additional Features As seen, the graph of quadratic functions are curves called parabolas. The curves will open upwards when a > 0 and have a minimum point and open downwards when a < 0 and have a maximum point. The maximum or minimum point or turning point is called the vertex.

43 It occurs when

44 Quadratic Models If the toy manufacturer finds the price as a function of demand to be find the equation for revenue and find the maximum revenue.

45 R(x) = Price  number of units equation for revenue Check: a = -1 < 0 so curve has maximum.

46 Maximum occurs when Maximum revenue is achieved when 5 units are produced.

47 If the costs of manufacture are $14 fixed and $1 per toy variable costs, determine the cost equation and any break-even points. cost equation

48 Break even points occur when

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50 Costs = revenue generated when 2 or 7 units of toys are produced.

51 Revenue Cost 27 $