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Splash Screen. Five-Minute Check (over Lesson 4–5) CCSS Then/Now New Vocabulary Key Concept: Quadratic Formula Example 1: Two Rational Roots Example 2: One Rational Root Example 3: Irrational Roots Example 4: Complex Roots Key Concept: Discriminant Example 5: Describe Roots

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–5) CCSS Then/Now New Vocabulary Key Concept: Quadratic Formula Example 1: Two Rational Roots Example 2: One Rational Root Example 3: Irrational Roots Example 4: Complex Roots Key Concept: Discriminant Example 5: Describe Roots Concept Summary: Solving Quadratic Equations Lesson Menu

  3. Solve x2 – 2x + 1 = 9 by using the Square Root Property. A. 4, 2 B. 4, –2 C. 2, –2 D. 2, 1 5-Minute Check 1

  4. A.–3, 3 B. C.–1, 1 D. Solve 4c2 + 12c + 9 = 7 by using the Square Root Property. 5-Minute Check 2

  5. A. B. C. D.1; (x + 1)2 Find the value of c that makes the trinomial x2 + x + c a perfect square. Then write the trinomial as a perfect square. 5-Minute Check 3

  6. A.–12, 12 B.–2, 2 C. D. Solve x2 + 2x + 24 = 0 by completing the square. 5-Minute Check 4

  7. A.2, –1 B. C. D.4, 2 Solve 5g2 – 8 = 6g by completing the square. 5-Minute Check 5

  8. Find the value(s) of k in x2 + kx + 100 = 0 that would make the left side of the equation a perfect square trinomial. A. –10, 10 B. 5, 20 C. –20, 20 D. 4, 25 5-Minute Check 6

  9. Content Standards N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. A.SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity. Mathematical Practices 8 Look for and express regularity in repeated reasoning. CCSS

  10. You solved equation by completing the square. • Solve quadratic equations by using the Quadratic Formula. • Use the discriminant to determine the number and type of roots of a quadratic equation. Then/Now

  11. Quadratic Formula • discriminant Vocabulary

  12. Concept

  13. ax2 + bx + c = 0 x2 – 8x = 33 1x2 – 8x – 33 = 0 Two Rational Roots Solve x2 – 8x = 33 by using the Quadratic Formula. First, write the equation in the form ax2 + bx + c = 0 and identify a, b, and c. Then, substitute these values into the Quadratic Formula. Quadratic Formula Example 1

  14. Two Rational Roots Replace a with 1, b with –8, and cwith –33. Simplify. Simplify. Example 1

  15. or Write as two equations. Two Rational Roots x = 11 x = –3 Simplify. Answer: The solutions are 11 and –3. Example 1

  16. Solve x2 + 13x = 30 by using the Quadratic Formula. A. 15, –2 B. 2, –15 C. 5, –6 D. –5, 6 Example 1

  17. One Rational Root Solve x2 – 34x + 289 = 0 by using the Quadratic Formula. Identify a, b, and c. Then, substitute these values into the Quadratic Formula. Quadratic Formula Replace a with 1, b with –34, and c with 289. Simplify. Example 2

  18. [–5, 25] scl: 1 by [–5, 15] scl: 1 One Rational Root Answer: The solution is 17. Check A graph of the related function shows that there is one solution at x = 17. Example 2

  19. Solve x2 – 22x + 121 = 0 by using the Quadratic Formula. A. 11 B. –11, 11 C. –11 D. 22 Example 2

  20. or or Irrational Roots Solve x2 – 6x + 2 = 0 by using the Quadratic Formula. Quadratic Formula Replace a with 1, b with –6, and c with 2. Simplify. Example 3

  21. [–10, 10] scl: 1 by [–10, 10] scl: 1 Irrational Roots Answer: Check Check these results by graphing the related quadratic function, y = x2 – 6x + 2. Using the ZERO function of a graphing calculator, the approximate zeros of the related function are 0.4 and 5.6. Example 3

  22. A. B. C. D. Solve x2 – 5x + 3 = 0 by using the Quadratic Formula. Example 3

  23. Complex Roots Solve x2 + 13 = 6x by using the Quadratic Formula. Quadratic Formula Replace a with 1, b with –6, and c with 13. Simplify. Simplify. Example 4

  24. [–5, 15] scl: 1 by [–5, 15] scl: 1 Complex Roots Answer: The solutions are the complex numbers 3 + 2iand 3 – 2i. A graph of the related function shows that the solutions are complex, but it cannot help you find them. Example 4

  25. ? (3 + 2i)2 + 13 = 6(3 + 2i)x = (3 + 2i) ? ? 9 + 12i + 4i2 + 13 = 18 + 12i Square of a sum; Distributive Property 22 + 12i – 4 = 18 + 12i Simplify. Complex Roots CheckTo check complex solutions, you must substitute them into the original equation. The check for 3 + 2i is shown below. x2 + 13 = 6x Original equation 18 + 12i = 18 + 12i  Example 4

  26. Solve x2 + 5 = 4x by using the Quadratic Formula. A. 2 ± i B. –2 ± i C. 2 + 2i D. –2 ± 2i Example 4

  27. Concept

  28. Describe Roots A.Findthevalueofthediscriminantforx2+3x + 5 = 0. Then describe the number and type of roots for the equation. a = 1, b = 3, c = 5 b2 – 4ac = (3)2 – 4(1)(5) Substitution = 9 – 20 Simplify. = –11 Subtract. Answer: The discriminant is negative, so there are two complex roots. Example 5

  29. Describe Roots B.Findthevalue of the discriminant forx2–11x+ 10 = 0. Then describe the number and type of roots for the equation. a = 1, b = –11, c = 10 b2 – 4ac = (–11)2 – 4(1)(10) Substitution = 121 – 40 Simplify. = 81 Subtract. Answer: The discriminant is 81, so there are two rational roots. Example 5

  30. A.Find the value of the discriminant for x2 + 8x + 16 = 0. Describe the number and type of roots for the equation. A. 0; 1 rational root B. 16; 2 rational roots C. 32; 2 irrational roots D. –64; 2 complex roots Example 5

  31. B.Find the value of the discriminant for x2 + 2x + 7 = 0. Describe the number and type of roots for the equation. A. 0; 1 rational root B. 36; 2 rational roots C. 32; 2 irrational roots D. –24; 2 complex roots Example 5

  32. Concept

  33. End of the Lesson

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