Chapter 7: Quadratic Equations Section 7.1: Solving Quadratic Equations by Factoring Terminology: Quadratic Equation: A polynomial equation of the second degree; the standard form of a basic equation is y = a 2 + b + c Where a, b, & c are real numbers and a 0. Review: Factoring Quadratic Equations (from Gr.10) Factoring A Quadratic Epression A quadratic equation can be solved in many cases by factoring. There are four major ways to factor a trinomial of the form y = a 2 + b + c. These were covered in math 1201 so we shall do a quick review. 1. Factoring Using Product and Sum: Product and Sum can only be used in situations where a=1. In such cases you must determine your factors by concluding what possible combination of two numbers can multiply to c and add to b EXAMPLES: Factor The following a. 2 + 6 + 8 b. k 2 7k 30 c. j 2 + 11j 42 d. f 2 9f + 20
2. Factoring Using Decomposition: Decomposition can be used in situations where a 1 and a GCF cannot be removed. In such cases you must determine your factors by following these steps: STEP1: Conclude what possible combination of two numbers can multiply to a c and add to b. STEP2: Decompose your middle term into those two numbers. STEP3: Group the first set and second set of terms. Pull out the GCF (Greatest Common Factor) of each group. STEP4: Then factor out the common bracketed term. EXAMPLES: Factor The following a. 5 2 7 6 b. 3k 2 13k 10 c. 8j 2 + 18j 5 d. 15f 2 7f 2 3. Factoring Using GCF: In some cases where a 1, a GCF can be removed from the situation and allow it to be factored using Product and Sum or via Decomposition (with slightly more manageable numbers). EXAMPLES: Solve The following a. 5 2 10 b. 3k 2 9k 12 c. 20 2 50 30 d. 10 2 + 14 + 12
4. Difference of Squares: Difference of squares can only be used in situations where b=0. In such cases both the a and c values will be perfect squares and there is a subtraction symbol between them. Your resulting factors will be the square root of each term with a different sign between them. EXAMPLES: Factor The following a. 2 9 b. 9k 2 25 c. 144j 2 100 d. 12f 2 75
Section 7.1: Solving Quadratic Equations by Factoring Terminology: Roots: The values of the variable that make an equation in standard form equal to zero. These are also called the solutions of the equation, the -intercepts or the zeros of a graph. These terms can and will be used interchangeably. NOTE: To determine the roots of an equation, we must make sure first that it equals zero. We then factor and set each factor equal to zero. This is called the zero product principal in which if the product of two real numbers is zero, then one or both of the numbers must be zero. E. Solve each equation by factoring. (a) 2 + 6 + 8 = 0 (b) k 2 + 8k + 7 = 0 (c) j 2 + 3j 54 = 0 (d) z 2 9z 70 = 0 (e) 5 2 7 6 = 0 (f) 4k 2 21k + 20 = 0 (g) 6j 2 + 18j = 0 (h) 63 2 56 = 0
(i) 2 9 = 0 (j) 144y 2 100 = 0 (k) 49h 2 64 = 0 (l) 45w 2 80 = 0 (m) 1 2 m2 + 3m + 4 = 0 (n) 1 5 p2 2p + 5 = 0 (o) 0.25q 2 + 0.5q 2 = 0 (p) 0.1h 2 1.2h 4.5 = 0
Word Problems Involving Factoring E. The entry to the main ehibit hall in an art gallery is a parabolic arch. The arch can be modelled by the function: h(w) = 0.625w 2 + 5w where the height, h(w), and width, w, are measured in feet. Several sculptures are going to be delivered to the ehibit hall in crates. Each crate is a square-based rectangular prism that is 7.5 ft high, including the wheels. The crates must be handled as shown, to avoid damaging the fragile contents. Determine the distance between the two points on the arch that are 7.5 ft high. E. Sanela sells posters to stores. The profit function for her business is: P(n) = 0.25n 2 + 6n 27 where n is the number of posters sold per month, in hundreds, and P(n) is the profit, in thousands of dollars. (ie if profit were $15 000, we would let P(n) = 15). (a) If Sanela wants to earn a profit of $5000 how many posters must she sell? (b) If Sanela wants to earn a profit of $9000, how many posters must she sell?
Determining a Possible Quadratic Equation Given its Roots E. Tori says she solved a quadratic equation by graphing it. She says that the roots were 5 and 7. Determine the quadratic equation that she may have solved. E. Write a quadratic function that has zeros at 0.5 and -0.75. E. The -intercepts of a quadratic function are 3 and 2.5. Write a quadratic equation that has these roots.
Section 7.2: Solving Quadratic Equations by Graphing Determining Solutions of a Quadratic Equation Determining the solutions from a graph is simple as you only need to identify the locations of the -intercepts taken directly from the graph E. Determine the roots of each quadratic function
Chapter 7: Quadratic Equations Section 7.3 Section 7.3: Using Quadratic Formula Terminology: Quadratic Formula: A formula for determining the roots of a quadratic equation in the form a 2 + b + c = 0, where a 0. The quadratic formula is written using the coefficients of the variables and the constant in the quadratic equation that is being solved: = b ± b2 4ac 2a Inadmissible Solution: A root of a quadratic equation that does not lead to a solution that satisfies the original problem. Solving Quadratic Equations Using the Quadratic Formula (a) 6 2 7 3 = 0 (b) 10 2 3 18 = 0
Chapter 7: Quadratic Equations Section 7.3 (c) 12 2 17 40 = 0 (d) 20p 2 + 7p + 3 = 0 (e) 1 2 z2 5z + 17 = 0
Chapter 7: Quadratic Equations Section 7.3 Word Problem Applications E. A student council is holding a raffle to raise money for a charity fund drive. The profit function for the raffle is: p(c) = 25c 2 + 500c 350 Where p(c) is the profit and c is the price of each ticket, both in dollars. (a) What ticket price would result in the student council raising $700 to donate? (b) What ticket price would result in the student council breaking even?
Chapter 7: Quadratic Equations Section 7.3 E. A rocket is launched into the air. Its motion can be modeled by the equation: h(t) = 5t 2 + 37t + 20 where h(t) is the height in metres above the ground and t is time in seconds. (a) Determine when the rocket first reaches a height of 80 metres? (b) How long does it take for the rocket to land back on the ground?
Chapter 7: Quadratic Equations Section 7.3 E. A photograph 8 cm by 11 cm will be framed as shown in the diagram. The combined area of the frame and photograph will be 180 cm 2. Algebraically determine the outside dimensions of the frame. 11 cm 8 cm E. A rectangular swimming pool has length 30 m and width 20 m. There is a deck of uniform width surrounding the pool. The area of the pool is the same as the area of the deck. Write a quadratic equation to model this situation and use it to determine the width of the deck. 30 m 20 m
Chapter 7: Quadratic Equations Section 7.3 E. A store rents an average of 750 video games each month at the current rate of $4.50. However, for every $1 increase, they know that they will rent 30 fewer games each month. Create a quadratic equation in standard form to represent the relationship and use it to determine the minimum number of times the owners should increase the rental rate enough to generate revenue of $5000 per month? E. The Math 2201 class is holding a bake sale to raise money for our class trip to PiWorld, the most eciting, most amazing, most interesting, most awesome Theme Park on the planet. They know that if they charge $7 per tray of cookies they will sell 400 trays. Their statistical analysis of the population has helped them to conclude that for every $1 they increase their price they will sell 30 less trays. Write a quadratic equation in standard form to represent this relationship. Use your equation to find out the minimum number of price increase they should make to reach their fundraising goal of $3200?