Solving Quadratic Equations by the Quadratic Formula Do Now: Solve for x by factoring first: x 2x 8 0 2 THE QUADRATIC FORMULA 1. When you solve using completing the square on the general formula ax 2 bx c 0 you get: b b2 4ac x 2a 2. This is the quadratic formula! Equation must be in standard form first-ax2+bx+c=0 3. Just identify a, b, and c then substitute into the formula. Do Using the Formula: List a, b, c first: x 2x 8 0 2 Do Using the Formula: List a, b, c first: x 2x 8 0 2 2 36 x 2 26 26 x 4, x 2 2 2 {4,2} Graph the following: Trace to the approximate roots: x 4x 10 0 2 Do Using the Formula: List a, b, c first: x 4x 10 0 2 Do Using the Formula: List a, b, c first: x 4x 10 0 2 4 56 x 2 4 56 2 2 14 x 4, x 2 14 2 1 Practice: Solve: 2x 4x 4 0 2 Practice: Solve: 2x 4x 4 0 2 4 (4) 4(2)(4) x 2(2) 2 4 48 4 4 3 x 1 3 4 4 WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it. An important piece of the quadratic formula is what’s under the radical: b2 – 4ac This piece is called the discriminant. WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical. WHAT THE DISCRIMINANT TELLS YOU! Value of the Discriminant Nature of the Solutions Negative 2 imaginary solutions Zero 1 Real Solution Positive – perfect square 2 Reals- Rational Positive – non-perfect square 2 Reals- Irrational Example Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1. 2 x 2 7 x 11 0 Discriminant = a=2, b=7, c=-11 b 2 4ac (7) 2 4(2)( 11) 49 88 Discriminant = 137 Value of discriminant=137 Positive-NON perfect square Nature of the Roots – 2 Reals - Irrational Students will review for the quadratics test. The discriminant-b2-4ac X2 -4x + 4 = 0 Disc = X2 -2x + 10 = 0 Disc= X2 +6x + 8 = 0 Disc = 2X2 -5x + 1 = 0 Disc = The discriminant X2 -4x + 4 = 0 Disc = 0 X2 -2x + 10 = 0 Disc= -36 X2 +6x + 8 = 0 Disc = 4 2X2 -5x + 1 = 0 Disc = 17 What is the nature of these roots? The discriminant X2 -4x + 4 = 0 X2 -2x + 10 = 0 X2 +6x + 8 = 0 2 2X -5x + 1 = 0 0, 1 real root Negative = (2 imaginary roots) pos perfect square= (2 rational roots) pos not perfect=2 irrational roots Do Now Solve for x: x 6x50 2 Solve for x: 2x 7x30 2 Now add and multiply the 2 roots: Solve for x: x 6x50 2 5,1 5 1 6 (5)(1) 5 Solve for x: 2x 7x30 2 1 3, 2 2x 7x30 2 1 3, 2 Sum and product of the roots: Sum of the roots: 1 7 3 2 2 Product of the roots: 1 3 (3)( ) 2 2 Sum and product of the roots: Sum of the roots b a Product of the roots: c a x 6x 8 0 2 Example: Sum = 6 b 6 c 8 , Product = 8 using sum and product: Do now: Hint: we know what the sum is: Given: X2 -2x + c = 0 and one root is 5 find the other root Find c using sum and product: Do now: Hint: we know what the sum is: Given: X2 -2x + c = 0 and one root is 1/3 find the other root using sum and product: Do now: Hint: we know what the product is: Given: X2 -bx + 10 = 0 and one root is 2, find the other root Use the discriminant For what value of c does the equation have equal roots: x 4x c 0 2 Use the discriminant For what value of c does the equation have equal roots: x 4x c 0 2 b 2 4ac 0 (4) 4(1)(c) 0 2 16 4c 0 16 4c c4 Use the discriminant For what value of c does the equation have imaginary roots: x 4x c 0 2 Use the discriminant For what value of c does the equation have imaginary roots: (less than zero) x 4x c 0 2 b 4ac 0 2 (4)2 4(1)(c) 0 16 4c 0 4c 16 c4 Quadratic equation knowing sum and product X2 – sum (x) +product = 0 Write the equation when the sum is - 6 and the product is 8 x 6x 8 0 2 Opposite of the sum, same as the product using sum and product: to write an equation Given the roots: 2, -5 Write the equation: Hint: we can find the sum and the product. using sum and product: to write an equation Given the roots: 2, -5 Write the equation: Sum= -3 Product = -10 x sum(x) product 0 2 x 3x 10 0 2 using sum and product: Do now: Given the roots: 2 + 6i,2-6i Write the equation: Hint: we can find the sum and the product. Quadratic equation knowing sum and product X2 – sum (x) +product = 0 Write the equation when the sum is 4 and the product is 40. x 4x 40 0 2 Quadratic equation knowing sum and product X2 – sum (x) +product = 0 Write the equation when the sum is 2/3 and the product is 4 Quadratic equation knowing sum and product X2 – sum (x) +product = 0 2 x x4 0 3 2 3x 2x 12 0 2 Write the equation when the sum is 2/3 and the product is 4 Clear the fraction! using sum and product: Do now: Given the roots: Write the equation: 2 3 2 Hint: we can find the sum and the product. using sum and product: Do now: Given the roots: Write the equation: 2 3 2 the sum is 2 and the product is 1/4. Quadratic equation knowing sum and product X2 – sum (x) +product = 0 Write the equation when the sum is 3/5 and the product 2/3 Clear the fraction! Quadratic equation knowing sum and product X2 – sum (x) +product = 0 3 2 x x 0 5 3 2 Write the equation when the sum is 3/5 and the product -2/3 Clear the fraction! Quadratic equation knowing sum and product X2 – sum (x) +product = 0 3 2 x x 0 5 3 2 15x 9x 10 0 2 Write the equation when the sum is 3/5 and the product -2/3 Clear the fraction! Example #1- continued Solve using the Quadratic Formula x 6x 28 0 2 a 1,b 6,c 8,b 4ac (6) 4(1)(8) 4 2 b b 2 4ac 2a 6 (6)2 4(1)(8) 2(1) 6 4 2 Reals - rational 2 62 62 4, 2 Example 2 Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1. 2 x 2 7 x 11 0 Discriminant = a=2, b=7, c=-11 b 2 4ac (7) 2 4(2)( 11) 49 88 Discriminant = 137 Value of discriminant=137 Positive-NON perfect square Nature of the Roots – 2 Reals - Irrational Example #2- continued Solve using the Quadratic Formula 2 x 2 7 x 11 0 a 2, b 7, c 11 b b 2 4ac 2a 7 7 2 4(2)(11) 2(2) 7 137 4 2 Reals - Irrational Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers. Find the sum and product of the roots: 1. x 2 2x 63 0 2. x 2 8x 84 0 3. x 2 5x 24 0 Sum Product Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers. Find the sum and product of the roots: 1. x 2 2x 63 0 2. x 2 8x 84 0 3. x 2 5x 24 0 Sum 1. -2 2. -8 3. 5 Product -63 -84 -24 Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers. Find roots: 1. x 2 2x 63 0 2. x 2 8x 84 0 3. x 2 5x 24 0 Roots: Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers. 1. x 2 2x 63 0 1. 9,7 2. x 2 8x 84 0 2.(6,14) 3. 3,8 3. x 2 5x 24 0 Review-Students will review for the quiz on quadratic equations Do Now: Find the nature of the roots for the following quadratic equations: 1. x 2 2x 6 0 2. x 2 8x 2 0 3. x 2 4x 5 0 Hint: use the discriminant. Solving Quadratic Equations by the Quadratic Formula Try the following examples. 1. x 2x 3 0 2 2. x 8x 12 0 2 3. x 4x 13 0 2 Roots: Solving Quadratic Equations using complete the Try the following example. Solve using complete the square: 1. x 2x 10 0 2

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