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9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Warm-Up Simplify the expression. 1. 3x +(– 6x) ANSWER –3 x 2. 5 + 4x + 2 ANSWER 4x + 7 3. 4(2x – 1) + x ANSWER 9x – 4 4. – (x + 4) – 6 x ANSWER – 7x – 4 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Warm-Up Simplify the expression. 11. (3xy)3 ANSWER 27x3y3 12. xy2 xy3 ANSWER x2 y5 13. (x5)3 ANSWER x15 ANSWER –x3 14. (– x)3 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Find the two square roots of each number. A. 49 – 49 = 7 49 = –7 B. 100 100 = 10 – 100 = –10 C. 225 – 7 is a square root, since 7 • 7 = 49. –7 is also a square root, since –7 • –7 = 49. 10 is a square root, since 10 • 10 = 100. –10 is also a square root, since –10 • –10 = 100. 225 = 15 15 is a square root, since 15 • 15 = 225. 225 = –15 –15 is also a square root, since –15 • –15 = 225. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Check It Out: Example 1 Find the two square roots of each number. A. 25 – 25 = 5 25 = –5 B. 144 144 = 12 5 is a square root, since 5 • 5 = 25. –5 is also a square root, since –5 • –5 = 25. 12 is a square root, since 12 • 12 = 144. – 144 = –12 –12 is also a square root, since –12 • –12 = 144. C. 289 289 = 17 – 17 is a square root, since 17 • 17 = 289. 289 = –17 –17 is also a square root, since –17 • –17 = 289. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Additional Example 3A: Evaluating Expressions Involving Square Roots Simplify the expression. 3 36 + 7 3 36 + 7 = 3(6) + 7 Evaluate the square root. = 18 + 7 Multiply. = 25 Add. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Additional Example 3B: Evaluating Expressions Involving Square Roots Simplify the expression. 25 + 3 16 4 25 + 3 = 16 4 3 1.5625 + 4 25 = 1.5625. 16 = 1.25 + 3 4 Evaluate the square roots. =2 Add. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Check It Out: Example 3A Simplify the expression. 2 25 + 4 2 25 + 4 = 2(5) + 4 Evaluate the square root. = 10 + 4 Multiply. = 14 Add. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Check It Out: Example 3B Simplify the expression. 18 + 1 t2 4 18 + 1 = 9 + 1 t2 4 4 18 = 9. t2 =3+ 1 4 Evaluate the square roots. = 31 4 Add. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Lesson Quiz for Student Response Systems 3. Evaluate the expression. A. 17 B. 17 C. 19 D. 72 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Lesson Quiz for Student Response Systems 4. Evaluate the expression. A. 4 B. 8 C. 16 D. 40 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. 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 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solving Quadratic Equations a. x2 = 4 b. x2 = 5 c. x2 = 0 d. x2 = -1 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solving Quadratic Equations a. x2 = 25 b. x2 = 7 c. x2 = 81 d. x2 = -12 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solve the equation. a. 2x2 = 8 SOLUTION a. 2x2 = 8 x2 = 4 x = ± 4 = ± 2 ANSWER Write original equation. Divide each side by 2. Take square roots of each side. Simplify. The solutions are – 2 and 2. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. EXAMPLE 1 b. m2 – 18 = – 18 m2 = 0 m= 0 ANSWER The solution is 0. Write original equation. Add 18 to each side.. The square root of 0 is 0. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. EXAMPLE 1 2 c. b + 12 = 5 Write original equation. b2 = – 7 Subtract 12 from each side. ANSWER Negative real numbers do not have real square roots. So, there is no solution. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. GUIDED PRACTICE Solve quadratic equations Solve the equation. 1. c2 – 25 = 0 SOLUTION c2 – 25 = 0 c = ± 25 = ± 5 ANSWER The solutions are – 5 and 5. Write original equation. Take square roots of each side. Simplify. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Take square roots of a fraction EXAMPLE 2 Solve 4z2 = 9. SOLUTION 4z2 = 9. Write original equation. 9 z2 = 4 z = ± Divide each side by 4. 9 4 z=± 3 2 ANSWER The solutions are – 3 and 3 2 2 Take square roots of each side. Simplify. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Approximate solutions of a quadratic equation Solve 3x2 – 11 = 7. Round the solutions to the nearest hundredth. SOLUTION 3x2 – 11 = 7 3x2 = 18 x2 = 6 x = ± 6 x ± 2.45 Write original equation. Add 11 to each side. Divide each side by 3. Take square roots of each side. Use a calculator. Round to the nearest hundredth. The solutions are about – 2.45 and about 2.45. 9.1 – Students will be able to evaluate square roots. Students will be able 1 equation by finding the square root. to EXAMPLE solve a quadratic Solve the equation. 2. 5w2 + 12 = – 8 Solve quadratic equations GUIDED PRACTICE SOLUTION 5w2 + 12 = – 8 5w2 = – 8 –12 w = –4 Write original equation. Subtract 12 from each side. Take square roots of each side. Simplify. ANSWER Negative real numbers do not have a real square root. So there is no solution. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solve the equation. 3. 2x2 + 11 = 11 Solve quadratic equations GUIDED PRACTICE SOLUTION 2x2 + 11 = 11 2x2 = 0 x=0 ANSWER The solution is 0 . Write original equation. Subtract 11 from each side. The root of 0 is 0. 9.1 – Students will be able to evaluate square roots. Students will be able 1 equation by finding the square root. to EXAMPLE solve a quadratic Solve quadratic equations GUIDED PRACTICE Solve the equation. 4. 25x2 = 16 SOLUTION 25x2 = 16 16 x = 25 x = ± 16 25 x=± 4 5 Write original equation. Divided each to by 25. Take square roots of each side. Simplify. ANSWER The solution is – 4 5 and 4 5 . 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solve quadratic equations Solve the equation. GUIDED PRACTICE 2 5. 9m = 100 SOLUTION 9m2 = 100 100 m= 9 m = ± 100 9 m = ± 10 3 Write original equation. Divided each to by 9. Take square roots of each side. Simplify. ANSWER The solution is – 10 and 10 . 3 3 9.1 – Students will be able to evaluate square roots. Students will be able 1 equation by finding the square root. to EXAMPLE solve a quadratic Solve quadratic equations Solve the equation. GUIDED PRACTICE 2 6. 49b + 64 = 0 SOLUTION 49b2 + 64 = 0 49b2 = – 64 –64 2 b = 49 b =– 64 49 ANSWER Write original equation. Subtract 64 from each side. Divided each to by 9. Take square roots of each side. Negative real numbers do have real square root. So there is no solution. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. GUIDED PRACTICE Solve quadratic equations Solve the equation. Round the solution to the nearest hundredth. 7. x2 + 4 = 14 SOLUTION x2 + 4 = 14 x2 = 10 Subtract 4 from each side. x = + – 10 Take square roots of each side. x = + – 3.16 ANSWER Write original equation. Use a calculation. Round to the nearest hundredth. The solutions are about – 3.16 and 3.16. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solve quadratic equations GUIDED PRACTICE Solve the equation. Round the solution to the nearest hundredth. 8. 3k2 – 1 = 0 SOLUTION 3k2 – 1 = 0 3k2 = 1 1 = 3 k = + – 1 3 k = + – 0.58 k2 Write original equation. 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. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. GUIDED PRACTICE Solve the equation. Round the solution to the nearest hundredth. 9. 2p2 – 7 = 2 SOLUTION 2p2 – 7 = 2 2p2 = 2 + 7 9 2 p = 2 p = + – 9 2 p = + – 2.12 Write original equation. 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. 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solving Quadratic Equations a. x2 + 5 = 21 b. x2 – 2 = 7 c. 2x2 = 18 d. 3x2 = 75 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solving Quadratic Equations a. 2x2 -8 = 0 b. x2 +25 = 0 c. x2 - 1.44 = 0 d. 5x2 = -15 9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Solving Quadratic Equations a. 3x2 -48 = 0 b. 120 - 6x2 = -30 c. 12x2 - 60 = 0