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7.5 Sum and Product of Roots

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7.5 Sum and Product of Roots. Algebra 2 Mrs. Spitz Spring 2007. Objectives:. Find the sum and product of the roots of a quadratic equation. Find all possible integral roots of a quadratic equation, and Find a quadratic equation to fit a given condition. Assignment. pp. 336-337 #7-41 odd.

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  1. 7.5 Sum and Product of Roots Algebra 2 Mrs. Spitz Spring 2007

  2. Objectives: • Find the sum and product of the roots of a quadratic equation. • Find all possible integral roots of a quadratic equation, and • Find a quadratic equation to fit a given condition.

  3. Assignment • pp. 336-337 #7-41 odd

  4. Introducing the lesson • There are times when you may know the roots of a quadratic equation but you don’t know the equation itself. For example, suppose the roots of quadratic equation are -7 and 5; and you want to find the equation. Remember when you used factoring to solve an equation. You eventually solved two equations set equal to zero to find the two solutions. You can use the process in reverse to find the equation when you know the solutions. (next slide)

  5. x = 5 - 5 - 5 x – 5 = 0 x = -7 + 7 + 7 x + 7 = 0 Working backwards (x – 5)(x + 7) = 0 Then FOIL x2 + 7x – 5x – 35 = 0 x2 + 2x = 35

  6. The sum and products are simply an extension of factoring. x2 + 2x – 35 = 0 1 -(-7 + 5)-7(5) 1 1 The last step • The last step shows the general form of a quadratic equation whose roots are 5 and -7. These roots, their sums and their product can lead you to the equation in another way. Study the pattern shown here:

  7. Sum and Product of Roots • The rule below can help you in checking your solutions to a quadratic equation or in finding the equation when you know the roots. If the roots of ax2 + bx + c with a ≠ 0 are s1 and s2, then:

  8. Ex. 1: Solve 3x2 – 16x – 12 = 0. Then use the sum and product of roots to check your solution.

  9. Ex. 1: Solve 3x2 – 16x – 12 = 0. Then use the sum and product of roots to check your solution.

  10. CHECK!!! {6, -⅔}--SUM 3x2 – 16x – 12 = 0 a = 3 b= -16 c = -12 The solution is correct according to the Sum of Roots Rule.

  11. CHECK!!! {6, -⅔}--PRODUCT 3x2 – 16x – 12 = 0 a = 3 b= -16 c = -12 The solution is correct according to the Product of Roots Rule

  12. Ex. 2: Write a quadratic equation that has roots of So, c = -80 ax2 + bx + c = 0 20x2 – 39x – 80 = 0 So, a = 20 and b = -39

  13. Ex. 3: Write a quadratic equation that has roots of 5 + 2i and 5 – 2i So, c = 29 ax2 + bx + c = 0 x2 – 10x + 29 = 0 So, a = 1 and b =-10

  14. Ex. 4: Find k such that -3 is a root of x2 + kx – 24 = 0 Let s1 = -3. Solve for s2. Then solve for k.

  15. Ex. 4: Find k such that -3 is a root of x2 + kx – 24 = 0 Check -3 is a root of x2 + kx – 24 = 0 x – 8 = 0 x = 8 It checks, so it’s correct.

  16. Reminders: • 7.6 will be completed by tomorrow. • Review for Chapter 7 is Monday. • We don’t meet Tuesday, Wednesday, and Thursday; so your test is scheduled for next Friday, March 2. • Third quarter is over March 14, which means that when my grades are due completed. • This means you must turn in everything to me no later than Wednesday, March 7.