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1. 3 x + (– 6 x )

–3 x. ANSWER. 4 x + 7. 9 x – 4. ANSWER. ANSWER. – 7 x – 4. ANSWER. Warm-Up. Simplify the expression. 1. 3 x + (– 6 x ). 2. 5 + 4 x + 2. 3. 4(2 x – 1) + x. 4. – ( x + 4) – 6 x. 27 x 3 y 3. x 2 y 5. x 15. – x 3. ANSWER. ANSWER. ANSWER. ANSWER.

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1. 3 x + (– 6 x )

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  1. –3 x ANSWER 4x + 7 9x – 4 ANSWER ANSWER –7x – 4 ANSWER Warm-Up Simplify the expression. 1. 3x +(– 6x) 2. 5 + 4x + 2 3. 4(2x – 1) +x 4. – (x + 4) – 6 x

  2. 27x3y3 x2y5 x15 –x3 ANSWER ANSWER ANSWER ANSWER 12.xy2xy3 Warm-Up Simplify the expression. 11. (3xy)3 13. (x5)3 14. (– x)3

  3. 49 = 7 49 = –7 – 100 = 10 100 = –10 – 225 = 15 225 = –15 – Find the two square roots of each number. A. 49 7 is a square root, since 7 • 7 = 49. –7 is also a square root, since –7 • –7 = 49. B. 100 10 is a square root, since 10 • 10 = 100. –10 is also a square root, since –10 • –10 = 100. C. 225 15 is a square root, since 15 • 15 = 225. –15 is also a square root, since –15 • –15 = 225.

  4. 25 = 5 25 = –5 – 144 = 12 144 = –12 – 289 = 17 289 = –17 – Check It Out: Example 1 Find the two square roots of each number. A. 25 5 is a square root, since 5 • 5 = 25. –5 is also a square root, since –5 • –5 = 25. B. 144 12 is a square root, since 12 • 12 = 144. –12 is also a square root, since –12 • –12 = 144. C. 289 17 is a square root, since 17 • 17 = 289. –17 is also a square root, since –17 • –17 = 289.

  5. 3 36 + 7 3 36 + 7 = 3(6) + 7 Additional Example 3A: Evaluating Expressions Involving Square Roots Simplify the expression. Evaluate the square root. = 18 + 7 Multiply. = 25 Add.

  6. 25 16 3 4 3 4 25 16 25 16 3 4 = 1.5625. 1.5625 + = + Additional Example 3B: Evaluating Expressions Involving Square Roots Simplify the expression. + 3 4 Evaluate the square roots. = 1.25 + = 2 Add.

  7. 2 25 + 4 2 25 + 4 = 2(5) + 4 Check It Out: Example 3A Simplify the expression. Evaluate the square root. = 10 + 4 Multiply. = 14 Add.

  8. 18t2 1 4 18t2 18t2 1 4 = 9. + Check It Out: Example 3B Simplify the expression. + 1 4 9 = + 1 4 = 3 + Evaluate the square roots. 1 4 = 3 Add.

  9. Lesson Quiz for Student Response Systems 3. Evaluate the expression. A. 17 B. 17 C.19 D.72

  10. Lesson Quiz for Student Response Systems 4. Evaluate the expression. A. 4 B. 8 C.16 D.40

  11. A Quadratic equation is an equation that can be written in the following standard form: ax2 + bx + c = 0 where a does not equal 0 If b = 0 ax2 + c = 0 These are the type we will work with today. If b = 0 and c = 0 ax2 = 0

  12. Solving Quadratic Equations • x2 = 4 • x2 = 5 • x2 = 0 • x2 = -1

  13. Solving Quadratic Equations • x2 = 25 • x2 = 7 • x2 = 81 • x2 = -12

  14. x = ± 4=±2  Solve the equation. a. 2x2 = 8 SOLUTION a.2x2 = 8 Write original equation. Divide each side by 2. x2 = 4 Take square roots of each side. Simplify. The solutions are–2and2. ANSWER

  15. ANSWER The solution is 0. EXAMPLE 1 b.m2 – 18 = – 18 Write original equation. m2 =0 Add 18 to each side.. m=0 The square root of 0 is 0.

  16. ANSWER Negative real numbers do not have real square roots. So, there is no solution. EXAMPLE 1 c.b2+12=5 Write original equation. b2 = – 7 Subtract 12 from each side.

  17. ANSWER The solutions are–5and5. c = ± 25=±5  GUIDED PRACTICE Solve quadratic equations Solve the equation. 1.c2 – 25 = 0 SOLUTION c2 – 25 = 0 Write original equation. Take square roots of each side. Simplify.

  18. ANSWER The solutions are – and 3 9 3 9 3 z2 = 4 2 4 2 2  z = ± z = ± Take square roots of a fraction Solve 4z2 = 9. EXAMPLE 2 SOLUTION 4z2 = 9. Write original equation. Divide each side by 4. Take square roots of each side. Simplify.

  19. 6 x = ± Approximate solutions of a quadratic equation Solve 3x2–11=7. Round the solutions to the nearest hundredth. SOLUTION 3x2–11=7 Write original equation. 3x2 = 18 Add 11 to each side. x2 = 6 Divide each side by 3. Take square roots of each side. x± 2.45 Use a calculator. Round to the nearest hundredth. The solutions are about –2.45 and about 2.45.

  20. ANSWER Negative real numbers do not have a real square root. So there is no solution.  w = –4 EXAMPLE 1 Solve quadratic equations Solve the equation. GUIDED PRACTICE 2. 5w2 + 12 = – 8 SOLUTION 5w2 + 12 = – 8 Write original equation. 5w2 = – 8 –12 Subtract 12 from each side. Take square roots of each side. Simplify.

  21. ANSWER The solution is 0. Solve quadratic equations Solve the equation. 3. 2x2 + 11 = 11 GUIDED PRACTICE SOLUTION 2x2 + 11 = 11 Write original equation. 2x2 = 0 Subtract 11 from each side. x = 0 The root of 0 is 0.

  22. 4 4 16 4 16 x = 5 5 5 25 25 x = ±  x = ± ANSWER The solution is – and . EXAMPLE 1 Solve quadratic equations Solve the equation. GUIDED PRACTICE 4. 25x2 = 16 SOLUTION 25x2 = 16 Write original equation. Divided each to by 25. Take square roots of each side. Simplify.

  23. 10 10 100 10 100 m = 3 3 3 9 9 m = ±  m = ± ANSWER The solution is – and . Solve quadratic equations Solve the equation. GUIDED PRACTICE 5. 9m2 = 100 SOLUTION 9m2 = 100 Write original equation. Divided each to by 9. Take square roots of each side. Simplify.

  24. 64 –64 49 49 b2= b = –  Negative real numbers do have real square root. So there is no solution. ANSWER EXAMPLE 1 Solve quadratic equations Solve the equation. GUIDED PRACTICE 6. 49b2 +64 = 0 SOLUTION 49b2 +64 = 0 Write original equation. 49b2= – 64 Subtract 64 from each side. Divided each to by 9. Take square roots of each side.

  25. + + – – 10 x = 3.16 x = The solutions are about – 3.16 and 3.16. ANSWER Solve quadratic equations GUIDED PRACTICE Solve the equation. Round the solution to the nearest hundredth. 7. x2 +4 = 14 SOLUTION x2 +4 = 14 Write original equation. x2= 10 Subtract 4 from each side. Take square roots of each side. Use a calculation. Round to the nearest hundredth.

  26. 1 1 3 3  + + – – k2= k = 0.58 k = Solve quadratic equations GUIDED PRACTICE Solve the equation. Round the solution to the nearest hundredth. 8. 3k2 –1 = 0 SOLUTION 3k2 –1 = 0 Write original equation. 3k2= 1 Add 1 to each side. Divided each to by 3. Take square roots of each side. Use a calculation. Round to the nearest hundredth. The solutions are about – 0.58 and 0.58.

  27. 9 9 2 2  + + – – p2= p = 2.12 p = GUIDED PRACTICE Solve the equation. Round the solution to the nearest hundredth. 9. 2p2 –7 = 2 SOLUTION 2p2 –7 = 2 Write original equation. 2p2= 2 + 7 Add 7 to each side. Divided each to by 2. Take square roots of each side. Use a calculation. Round to the nearest hundredth. The solutions are about – 2.12 and 2.12.

  28. Solving Quadratic Equations • x2 + 5 = 21 • x2 – 2 = 7 • 2x2 = 18 • 3x2 = 75

  29. Solving Quadratic Equations • 2x2 -8 = 0 • x2 +25 = 0 • x2 - 1.44 = 0 • 5x2 = -15

  30. Solving Quadratic Equations • 3x2 -48 = 0 • 120 - 6x2 = -30 • 12x2 - 60 = 0

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