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10.7

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10.7

  1. 1. Derive the Quadratic Formula ax 2 + bx + c = 0
  2. 2. The Quadratic Formula The roots of the polynomial ax 2 + bx + c and the solutions of the quadratic equation ax 2 + bx + c = 0 -b ± b2 - 4ac are x = where a ¹ 0 and b2 - 4ac ³ 0 2a
  3. 3. Example 1 Find the roots of a polynomialFind the roots of x2 – 6x + 3.SOLUTIONThe roots of x2 – 6x + 3 are the values of x for whichx2 – 6x + 3 = 0. –b + – b2 – 4acx= Quadratic formula 2a – ( – 6) + – ( – 6)2 – 4( 1 )( 3 ) Substitute values in the quadraticx= formula: a = 1, b = – 6, and c = 3. 2( 1 )
  4. 4. Example 1 Find the roots of a polynomial 6 + 24 = – Simplify. 2 6 +2 6 = – Simplify radical. 2 2(3 + 6 ) – = = 3 + – 6 Divide out factor of 2. 2ANSWERThe roots of x2 – 6x + 3 are 3 + 6 and 3 – 6.
  5. 5. Example 1 Find the roots of a polynomialCHECK Substitute each root for x. The polynomial should simplify to 0. (3 + 6 )2 – 6 (3 + 6) + 3 = 9 + 6 6 + 6 – 18 – 6 6 + 3 = 0 (3 – 6 )2 – 6 (3 – 6) + 3 = 9 – 6 6 + 6 – 18 + 6 6 + 3 = 0
  6. 6. Example 2 Multiple Choice PracticeWhich is one of the solutions to the equation2x2 – 7 = x? 1 1 – 57 + 57 4 4 –1 + 57 1+ 57 4 4SOLUTION Write original 2x2 – 7 = x equation. Write in standard2x2 – x – 7 = 0 form.
  7. 7. Example 2 Multiple Choice Practice –b+ – b2 – 4ac Quadratic formula x = 2a Substitute values – ( – 1) + ( – 1)2 – 4( 2 )(–7) in the quadratic – formula: = 2( 2 ) a = 2, b = –1, and c = –7. 1 + 57 – = Simplify. 4ANSWER 1 + 57One solution is . 4The correct answer is D.
  8. 8. Methods for Solving Quadratic Equations
  9. 9. Example 4 Choose a solution methodTell what method(s) you would use to solve the quadraticequation. Explain your choice(s).a. 10x2 – 7 = 0b. x2 + 4x = 0c. 5x2 + 9x – 4 = 0 SOLUTIONa. The quadratic equation can be solved using square roots because the equation can be written in the form x2 d. =
  10. 10. Example 4 Choose a solution methodb. The quadratic equation can be solved by factoring because the expression x2 + can be factored easily. 4x Also, the equation can be solved by completing the square because the equation is of the form where a ax2and b is an=even number. 1 + bx + c 0 =c. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation should be solved using the quadratic formula.
  11. 11. Example 3 Use the quadratic formulaFILM PRODUCTIONFor the period 1971 – 2001, the number y of filmsproduced in the world can be modeled by the functiony = 10x2 – 94x + 3900 where x is the number of yearssince 1971. In what year were 4200 films produced?SOLUTION y = 10x2 – 94x + 3900 Write function. 4200 = 10x2 – 94x + 3900 Substitute 4200 for y. 0 = 10x2 – 94x – 300 Write in standard form.
  12. 12. Example 3 Use the quadratic formula Substitute values in the – ( – 94) + – ( – 94)2 – 4 (10)(–300) quadratic formula: a = 10,x = 2(10) b = –94, and c = –300. 94 + 20,836 – Simplify. = 20 94 + 20,836The solutions of the equation are ≈ 12 and 2094 – 20,836 ≈ –3. 20ANSWERThere were 4200 films produced about 12 years after 1971,or in 1983.
  13. 13. 10.7 Warm-up (Day 1)Use the quadratic formula to Find the roots1. x 2 + 4x -1Use the quadratic formula to solve the equation2. x 2 - 8x +16 = 03. x 2 - 5x = -214. 4z 2 = 7z + 2
  14. 14. 10.7 Warm-up (Day 2)Use the quadratic formula to solve the equation1. n2 +1 = 5n2. 2z + 4 = 3z 2
  15. 15. 10.7 Warm-up (Day 3)Use the quadratic formula to solve the equation1. 4x 2 + 6x = 3x 2 - 4x +12. 7r 2 - 2r = 6

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