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Quadratic Equations Irrational Roots and Sum & Product of the Roots.

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Presentation on theme: "Quadratic Equations Irrational Roots and Sum & Product of the Roots."— Presentation transcript:

1 Quadratic Equations Irrational Roots and Sum & Product of the Roots

2 Solve for x by using the quadratic formula:
Aim: Use the sum and product of the roots in order to write a quadratic equation. Solve for x by using the quadratic formula: a. What is the sum of its roots? = -7 b. What is the product of its roots? -3  -4 = 12

3 Sometimes you are given information backwards!
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sometimes you are given information backwards! You may be given information and asked to write a quadratic equation. Remember – it’s all about finding which factors of ac add to b.

4 Sum and Product of Roots for a Quadratic Equation
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum and Product of Roots for a Quadratic Equation The sum of the roots is:

5 The product of the roots is:
Aim: Use the sum and product of the roots in order to write a quadratic equation. The product of the roots is:

6 Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum of roots = Product of roots = Given the equation x2 + x – 20 = 0, a = 1 b = 1 c = -20 a. What is the sum of the roots? = = -1 b. What is the product of the roots? = = -20

7 Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum of roots = Product of roots = The roots of a quadratic are and -b = 39 a. What is the sum of the roots? b = -39 a = 20 b. What is the product of the roots? c = - 80 c. What is the quadratic equation? 20 x2 -39 x - 80 = y

8 Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum of roots = Product of roots = The roots of a quadratic are 5 + 2i and 5 – 2i. a. What is the sum of the roots? -b = 10 b = - 10 (5 + 2i) + (5 – 2i) = 10 a = 1 b. What is the product of the roots? (5 + 2i)  (5 – 2i) = i – 10i – 4i2 c = + 29 25 – 4(-1) = 29 c. What is the quadratic equation? 1 x2 - 10 x + 29 = y

9 Irrational Root Theorem
Aim: Use the sum and product of the roots in order to write a quadratic equation. Irrational Root Theorem For a polynomial 𝑦= 𝑎 0 𝑥 𝑛 + 𝑎 1 𝑥 𝑛−1 +…+ 𝑎 𝑛−1 𝑥+ 𝑎 𝑛 If 𝑎 + 𝑏 is a root, Then 𝑎 − 𝑏 is also a root Irrationals always come in pairs. Real values do not. Conjugate: Complex pairs in form 𝑎+ 𝑏 and 𝑎− 𝑏

10 Example of Irrational Root Theorem
Aim: Use the sum and product of the roots in order to write a quadratic equation. Example of Irrational Root Theorem 1. A polynomial function with roots 𝟐+ 𝟑 . Find all additional roots and degree of polynomial. 𝟐− 𝟑 2 Other root:________ Degree of Polynomial:________ 2. A polynomial function with roots −𝟏, 𝟎, − 𝟑 , 𝟏− 𝟓 . Find all additional roots and degree of polynomial. 𝟑 , 𝟏+ 𝟓 6 Other root:________ Degree of Polynomial:________ 3. A polynomial function with roots 𝟏− 𝟑 , 𝟐+𝒊. Find all additional roots and degree of polynomials. 1+ 𝟑 , 𝟐−𝒊 4 Other root:________ Degree of Polynomial:________

11 Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum of roots = Product of roots = One root of a quadratic is a. What is the sum of the roots? -b = 6 b = - 6 a = 1 b. What is the product of the roots? c = + 7 c. What is the quadratic equation? 1 x2 - 6 x + 7 = y

12 Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum of roots = Product of roots = The sum of the roots of a quadratic is , the product of the roots is What is the equation of the quadratic? -b = 3 c = -2 b = - 3 a = 4 a = 2 ( ) 2 a = 4 4 x2 - 3 x - 2 = y

13 Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation. Sum of roots = Product of roots = Find k such that -3 is a root of x2 + kx – 24 = 0. Product = Sum =

14 Polynomials equations greater than quadratics.
Aim: Use the sum and product of the roots in order to write a quadratic equation. Polynomials equations greater than quadratics. Write a polynomial function of least degree that has the given zeros. Roots:______ 5, − 3 Other possible roots:______ 3 Degree of Polynomial:________ 3 Then 𝑥=5, 𝑥=− 3 𝑎𝑛𝑑 𝑥= 3 So, 𝑥−5=0, 𝑥+ 3 =0 𝑎𝑛𝑑 𝑥− 3 =0

15 If 𝑥−5=0, 𝑥+ 3 =0 𝑎𝑛𝑑 𝑥− 3 =0 then (𝑥−5)(𝑥+ 3 )(𝑥− 3 )=0 Foil
Aim: Use the sum and product of the roots in order to write a quadratic equation. If 𝑥−5=0, 𝑥+ 3 =0 𝑎𝑛𝑑 𝑥− 3 =0 then (𝑥−5)(𝑥+ 3 )(𝑥− 3 )=0 Foil (𝑥−5)( 𝑥 2 −3)=0 Foil again or distrubute 𝑥 3 −5 𝑥 2 −3𝑥+15=0 If needed distribute again or set equal to 𝑦 𝑥 3 −5 𝑥 2 −3𝑥+15=𝑦

16 Other possible roots:________ − 5 , 2
Aim: Use the sum and product of the roots in order to write a quadratic equation. Write a polynomial function of least degree that has the given zeros: 5 , − 2 Other possible roots:________ − 5 , 2 Degree of Polynomial:________ 4 𝑥= 5 , 𝑥=− 5 , 𝑥=− 2 𝑎𝑛𝑑 𝑥= 2 𝑥− 5 =0, 𝑥+ 5 =0, 𝑥+ 2 =0 𝑎𝑛𝑑 (𝑥− 2 )=0 𝑥− 5 𝑥 𝑥+ 2 (𝑥− 2 )=0 𝑥 2 −5 ( 𝑥 2 −2)=0 ( 𝑥 4 −7 𝑥 2 +10)=0 𝑥 4 −7 𝑥 2 +10=𝑦


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