Applications of Quadratic Equations Topic 6 Applications of Quadratic Equations Unit 7 Topic 6

Information A quadratic equation can be solved in a variety of ways. Factoring Some quadratic equations can be solved by factoring. To factor an equation, start by writing the equation in standard form. Factor using the common factor, a difference of squares or a trinomial method. Set each factor equal to zero and solve the resulting linear equations. Each root is a solution to the original equation.

Quadratic Formula •The roots of a quadratic equation in standard form, 𝑎𝑥 2 +𝑏𝑥+𝑐=0 , where a ≠ 0, can be determined by using the quadratic form 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 . • A quadratic equation can have one distinct real root, two distinct real roots or no real roots.

Graphing •Method 1: Find the x-Intercepts of the Corresponding Function You can solve quadratic equations of the form , by graphing the corresponding quadratic function, , and finding the x-intercepts of the graph of . • Method 2: Find the x-Coordinates at the Points of Intersection of a System of Equations You can solve quadratic equations of the form , by graphing the corresponding quadratic functions, and , and then finding the x-coordinate at each point of intersection of and . When you use a quadratic equation to solve a problem, you need to verify that each of the roots make sense in the context of the situation.

Try this on your own first!!!! Example 1 Try this on your own first!!!! Solving a Problem Using Factoring The manager of a fashion store is investigating how raising or lowering the price of a purse changes the daily revenue from the purse. The function gives the store’s revenue, R, in dollars, from purse sales, where x is the price change, in dollars. a) Write a quadratic equation that you could use to find what price changes will result in no revenue?

Example 1a: Solution Write a quadratic equation that you could use to find what price changes will result in no revenue?

Example 1 Solving a Problem Using Factoring b) Use factoring to determine what price changes will result in no revenue. GCF of -1 Product = -100 Sum = -15 5, -20 A decrease by $5 or an increase by $20 will result in no revenue.

Example 2 Try this on your own first!!!! Solving a Problem Using the Quadratic Formula Eric makes an open-topped box from a piece of cardboard measuring 14 cm by 10 cm. He forms the sides of the box by cutting out squares of x centimetres from each corner of the rectangular sheet, as shown in the diagram below.

Example 2 Try this on your own first!!!! Eric wants the area of the base of the box to be 45 cm2. Eric lets x represent the side length of the square cut from each corner. Into the area formula for a rectangle, , Eric substituted A = 45, and . Finish Eric’s solution. a) Express the equation in standard form, .

Example 2a: Solution Express the equation in standard form, .

Example 2b: Solution Use the quadratic formula to solve for x. a b c a = +4 b = -48 c = +95 You have to enter this in your calculator twice. Once with the positive and once with the negative.

Example 2c: Solution What are the dimensions of the box? It is not possible to cut out 9.5 cm from each end of a 14 cm sheet. The only possible dimensions of the box, such that the area of the base is 45 cm2, are a length of 9 cm, a width of 5 cm, and a height of 2.5 cm. Cut out squares with side length of 2.5 cm (equal to the height of the box) in each corner of the cardboard.

Example 3 Try this on your own first!!!! Solving a Problem Using Graphing A football kicker kicks or punts a football into the air. The path of the ball can be modelled by the function , where h is the height of the football, in feet, and d is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot. d h

Example 3 Try this on your own first!!!! Solving a Problem Using Graphing a) Suppose the nearest defensive player jumps up to a height of 9 feet and attempts to block the punt, as shown in the diagram above. Graphically determine how far the defensive player is from the point of impact. Answer to the nearest tenth of a foot.

Horizontal distance (ft) Example 3a: Solution Suppose the nearest defensive player jumps up to a height of 9 feet and attempts to block the punt, as shown in the diagram above. Graphically determine how far the defensive player is from the point of impact. Answer to the nearest tenth of a foot. Horizontal distance (ft) x Height (ft) y 2nd trace 5: intersect Possible Window Settings The defensive player is 6.8 feet from the point of impact.

Example 3b: Solution Suppose the ball is not blocked by the defensive player or any other player. Graphically determine how far down the field the ball will go when it hits the ground, rounded to the nearest tenth of a foot. 2nd trace 2: zero Possible Window Settings The ball will have travelled about 111.8 feet when it hits the ground.

Need to Know: There are many real-life applications that can be represented by quadratic equations. A quadratic equation can be solved in a variety of ways, such as; graphically or using the quadratic formula. Some quadratic equations can be solved by factoring. When you solve a quadratic equation, verify that the roots make sense in the context of the problem. You’re ready! Try the homework from this section.