THE QUADRATIC QUANDARY By Jay Snyder jaysnydermathplus@comcast.net I II III IV V What is a Quadratic Equation? What is a Quadratic Used For? Solve the Quadratic by Factoring Quadratic Formula Application
I What is a Quadratic Equation? quad means square. The eponent is (squared). The equation has degree. Standard Form (generic form for parabola): a + b + c = 0 where a, b & c are numbers and aǂ0. Identify the quadratic : Linear Equation : 3 17 Graph is a line Quadratic Equation : 3-14 0 Graph is a parabola Cubic Equation : 4 3 3 0 Graph is a "squiggle" Eamples of quadratics: Parabola: 8 1 0 Circle : y 9 Ellipse: 9 y 4 1 Hyperbola : 16 y 9 1
II What is a Quadratic Used For? Quadratics are widely used in science, business, and engineering. 1. The U-shape of a parabola can describe the trajectories of an object when it flies through the air. From the equation we can find the height of an object after a given length of time or tell how long it will take for an object to reach a certain height. The military can predict when or where artillery will hit the earth or target.. Quadratics can be used to design parabolic shapes such as satellite dishes and car headlights. Most of the objects we use every day were created by applying a quadratic function. 3. The shape of cables in a suspension bridge can be determined with the quadratic. 4. Quadratics are used to help forecast business profit & loss and assist in determining minimum and maimum values. 5. Quadratic equations are used in situations where two things are multiplied together and they both depend of the same variable. For eample, when working with area, if both dimensions are written in terms of the same variable, a quadratic equation can be used. 6. The quadratic can be used to eplain how planets in our solar system revolve (elliptically) around the sun. Gravity Sales Area of Frame Best price area height profit width time price
III Solve the Quadratic by Factoring A B Factoring 1 GCF Riddle Game 3 Difference of Squares 4 Factoring Combinations 5 Grouping & Riddle Game Epanded [jmath.net/algebra 1 course/video #17] Solve by Factoring 1 The Zero Guarantee Solve the Quadratic Equation with 3 Steps a) Set Equation Equal to Zero b) Factor c) Use the Zero Guarantee Factoring GCF (Greatest Common Factor) 1) Define factor as a noun: numbers multiplied together to get a given product. Factors of 1: 1 1, 6, 3 4 {1,, 3, 4, 6, 1} Factors of 18: 1 18, 9, 3 6 {1,, 3, 6, 9, 18} ) CHANGE THE MINDSET FOR ALGEBRA! What numbers can you divide by? These are the factors. What numbers can you divide into 1? 1,, 3, 4, 6, 1 NOTE: If you can divide by a number, it must have another factor. 1 is divisible by 6. Therefore, there is another factor (which is ). 3) GCF of 1 & 18 Common Factors of 1 & 18: 1 3 4 6 1 1 3 6 9 18 GCF The GCF of 1 & 18 is 6. (The largest number that will divide into 1 & 18)
4) Prime Factorization (factor tree) Prime numbers have eactly factors. 7 is a prime number. Its factors are 1 & 7. 19 is a prime number. Its factors are 1 & 19. Prime factorization breaks a number down using its factors until all the factors are prime. 1 6 3 3 = 1 OR 3 1 3 4 3 = 1 OR 3 5) Define factor as a verb: to epress as a product of or more quantities. We factor to change addition or subtraction to multiplication! Arithmetic: 10 + 15 The GCF of 10 & 15 is 5. Use the GCF (5) and factor (pull out the GCF) and change to multiplication. 5( + 3) The addition (10+15) has been changed to multiplication 5(+3)! Algebra: 10 + 15 GCF = 5 5( + 3) Addition changed to multiplication! Algebra: 9 + 1 To find GCF, find GCF of the numbers (coefficients) first (3) Then find the GCF of the variables (). GCF = 3 3(3 + 4) Addition changed to multiplication!
Riddle Game Eample 1: I m thinking of numbers whose product is 1 and whose sum is 8. What are the numbers? Answer: and 6 Eample : I m thinking of numbers whose product is 4 and whose sum is 11. What are the numbers? Answer: 3 and 8 Eample 3: I m thinking of numbers whose product is -15 and whose sum is. What are the numbers? Answer: 5 and -3 Eample 4: I m thinking of numbers whose product is 1 and whose sum is -7. What are the numbers? Answer: -3 and -4 Review Distributive Property: 3( + 4) = 3 + 1 ( +) = + ( + 3)( + ) = + + 3 + 6 = + 5 + 6 To factor a quadratic trinomial, play the riddle game. Eample 1: + + 5 + 6 Riddle Game: Multiply to get 6 and add to get 5. Answer: and 3 ( + )( + 3) Answer to riddle NOTE: the addition has been changed to multiplication. + Eample : Factor + 9 + 14 Riddle Game: Multiply to get 14 and add to get 9. Answer to riddle: and 7 Answer: ( + )( + 7) ***The riddle game will not work unless the coefficient is 1. 3 + 4 6 will not factor with the riddle game because of the coefficient 3.
Difference of Squares Difference refers to subtraction. A square number is the product of a number times itself. Eamples of square numbers: 1, 4, 9, 16, 5, 36, 49, 64,, 9, Look at this multiplication: ( - 4)( + 4) = + 4 4 16 = - 16 Notice 4 4 cancels out the term. The final answer is a difference of squares (the and 4 ). Therefore, factoring the difference of squares is one of the easiest epressions to factor (as long as you recognize it! It needs to JUMP OFF THE PAGE AT YOU!!!) It s easier than finding the GCF or using the riddle game. Eamples: 9 ( 3)( 3) 4 ( )( ) 49 ( 7)( 7) 64 ( 8)( 8) 9 4y (3 y)(3 y) 4 6 5 ( 3 5)( 5)
Factoring Combinations Always look for the GCF before other factoring tools. Eample 1: Factor 8 3 8( 4) (GCF = 8) 8( + 4)( 4) (Difference of Squares) Eample : Factor 3 + 3 18 3( + 6) (GCF = 3) 3( + 3)( ) (Riddle Game) Solve by Factoring Eample 3: Factor 4 ( 4 1) (GCF = ) The Zero Guarantee (Zero Product Property) ( + 1)( - 1) (Difference of Squares) ( + 1)( + 1)( 1) (Difference of Squares) If a b = 0, what conclusion can you make about a or b? a or b must equal 0. THIS SIMPLE CONCEPT IS USED TO SOLVE QUADRATICS! To use this concept, we must have a multiplication problem. That s why we factor. It changes addition or subtraction to multiplication. If 8 = 0, then must equal 0. If ( )( 7) = 0, then = OR = 7 If ( + 3)( 1) = 0, then = -3 OR = 1 If ( + 4)( + 5) = 0, then = 0, = -4 OR = -5
Solve the Quadratic Equation with 3 Steps Eample 1: Solve the Quadratic + 7 + 10 = 0 Step 1: Make sure the equation is set equal to zero. The Zero Guarantee will not work if the equation does not equal zero. This equation is already set equal to zero. Step : Factor the equation. ( + )( + 5) = 0 (Riddle Game) Step 3: Use the Zero Guarantee to solve. Solution to equation: {-,-5} Eample : Solve the Quadratic - 4-1 = 0 Step 1: The equation is already set zero. Step : Factor the equation. ( 6)( + ) = 0 (Riddle Game) Step 3: Use the Zero Guarantee to solve. Solution to equation: {-,6} Eample 3: Solve the Quadratic 5 45 = 0 Step 1: The equation is already set equal to zero. Step : Factor the equation. 5( 9) = 0 (GCF) 5( + 3)( 3) = 0 (Difference of Squares) Step 3: Use the Zero Guarantee to solve. Solution to equation: {-3,3}
Eample 4: Solve for. + 18 = -16 Step 1: Set the equation equal to zero. Add 16 to both sides of equation. + 18 + 16 = 0 Step : Factor the equation. ( + 9 + 8) = 0 (GCF) ( + 1)( + 8) = 0 (Riddle Game) Step 3: Use the Zero Guarantee to solve. Solution to equation: {-1,-8} IV Quadratic Formula If you can t or don t want to factor a quadratic equation, you can solve it with the Quadratic Formula. The Quadratic Formula looks a little intimidating. But with a closer look, it s not! The Quadratic Formula: b b 4ac a Identify a, b & c NOTE: There are a s, b s and 1 c. (Highlighting is helpful!) All we need to do is substitute the numbers for a, b & c and then crunch the numbers! Standard form of the quadratic (parabola) is: a + b + c = 0 where a, b and c are the numbers we need for the Quadratic Formula. Identify a, b and c in the following eamples. Eample 1: 4 +7 + 3 = 0 a = 4 b = 7 c = 3 Eample : + - 5 = 0 a = 1 b = c = -5 Eample 3: 6 - + 8 = 0 a = 6 b = -1 c = 8 Eample 4: 3 + 6 = 0 a = 3 b = 6 c = 0 Eample 5: 5-6 = 0 a = 5 b = 0 c = -6
Evaluate the Formula 1) Eplain the symbol. It is a shortcut for writing two problems at one time. 8 means 8 + and 8 OR 10 and 6. ) Solve the same equation by factoring and then by using the Quadratic Formula. Solve for by factoring 5 6 0 ( )( 3) 0 Solution :{, 3}} Solve for with the Quadratic Formula 5 6 0 a 1, b 5, c 6 b b 4ac a Plug in a, b and c : 5 5 5 4(1)(6) (1) 5 4 5 1 5 1 5 1, 4 6,, 3 Solution :{, 3} As you can see, factoring to solve is much less work than using the Quadratic Formula. Always try to factor first!
3) Solve using the Quadratic Formula. (this equation does not factor) 3 7 3 0 a 3, b 7, c 3 b b 4ac a Plug in a, b and c : 7 7 7 6 7 4(3)(3) (3) 49 36 6 13 7 Solution : 6 13 7, 6 13 Use of the Discriminant The discriminant is the epression under the radical sign in the Quadratic Formula: b 4ac Evaluating the discriminant will tell you how many solutions the equation has. It does not tell you what the solutions are! A discriminant greater than zero means there will be solutions to the equation. A discriminant that equals zero means there will be 1 solution to the equation. A discriminant less than zero means there are no solutions to the equation. Look at these 3 possible solutions: The discriminant is 5. The discriminant is 0. The discriminant is -7. There are solutions. There is 1 solution. There are no real solutions. 15 7 5 3 7 0 7 9
V Application The quadratic equation h o = -16t + vt + h i is used with gravity problems. With this equation we can find the time an object will reach a given height or find the height of an object after a given amount of time. Eample 1: h o = the height of the object at a given time (t) t = the time in seconds that the object travels v = the initial velocity of the object in feet per second h i = the initial height of the object in feet Carol dove into a swimming pool from a 15 foot high diving board. Her initial upward velocity was 8 feet per second. Find the time (t) in seconds it took Carol to enter the water. Substitute the values from the problem into the equation h o = -16t + vt + h i h o = 0 v = 8 h i = 15 The height of Carol when she enters the water The initial velocity 8 feet per second Carol s initial height from a 15 foot diving board 0 = -16t + 8t + 15 Solve for t using the Quadratic Formula. a = -16 b = 8 c = 15 8 t 8 4( 16)(15) ( 16) 8 t t 4 3 t.75, 104 3 40, 3 1.5 8 3 3 The time (t) that it took Carol to enter the water cannot be negative. Therefore, it took her 1.5 seconds to enter the water.
Eample : Using the previous problem, what was Carol s height after 1 second? Substitute 1 for t and solve for h o. h o = -16(1) + 8(1) + 15 h o = 7 After 1 second Carol was 7 feet above the water. Eample 3: Tom threw a baseball in the air from a height of 6 feet with an initial upward velocity of 14 feet per second. James caught the ball on its way down at a point 4 feet above the ground. How long was the ball in the air before James caught it? Substitute the values from the problem into the equation h o = -16t + vt + h i h o = 4 v = 14 h i = 6 The height of the ball when it was caught The initial velocity 14 feet per second Height of ball when it left Tom s hand 4 = -16t + 14t + 6 Solve for t using the Quadratic Formula Set the equation equal to zero (subtract 4 from both sides of equation) 0 = -16t + 14t + Identify a, b and c a = -16 b = 14 c = t 14 14 4( 16)() ( 16) 14 34 t 3 4 3 t, 3 3 t.15, 1 14 18 3 The time (t) the ball was in the air cannot be negative. Therefore, the ball took 1 second to go from Tom to Jerry.