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Quadratic Equations: Factoring, Completing the Square, and Vertex Form

Learn how to solve quadratic equations by factoring, completing the square, and converting to vertex form. Practice examples included for better understanding.

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Quadratic Equations: Factoring, Completing the Square, and Vertex Form

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  1. Warm-Up • Solve by factoring: • Write the quadratic equation in standard form with the roots: Find the value of c that makes each trinomial a perfect square: 3) 4) Write the perfect square for each: 5) 6)

  2. Warm-Up Solve by factoring: 5. x2 +18x+81 = 0 6. 3x2 + 8x = 3x + 2 • Solve: • (x – 5)2=49 • 3x2 + 8 =-76 • Simplify: • Simplify:

  3. Multiply: Warm-Up 3. (x + 3)2 4. (x – 5)2 • (x + 7)2 • (3x+2)2 • (4x – 6)2 Simplify: 1. 2.

  4. 1.7 Completing the square Solve by finding square roots 1. x2 + 10x + 25 = 49 (x + 5)2 = 49 x + 5 = + 7 x = -5 + 7 x = -12 x = 2 {-12, 2}

  5. continued • x2 – 6x + 9 = 32 (x – 3)2 = 32 x – 3 = + x – 3 = x = 3 + 4 x ≈ 8.7 x ≈ -2.7

  6. Complete the square 3. x2 + 8x – 20 = 0 x2 + 8x + ___ = 20 + ___ x2 + 8x + 42 = 20 + 16 (x + 4)2 = 36 x + 4 = +6 x = -4 + 6 x = -10, x = 2 {-10, 2}

  7. continued 4. x2 + 4x + 11 = 0 x2 + 4x + _____ = -11 + ___ x2 + 4x + 22 = -11 + 4 (x + 2)2 = -7 x + 2 = x =

  8. Your Turn • x2 + 14x – 15 = 0 • x2 – 10x +13 = 0 5. {-15,1}

  9. Warm-Up Solve by using square roots: • x2 – 16x +64 = 27 Solve by completing the square: 2. x2 + 4x – 12 = 0 3. x2 + 2x + 3 = 0 {-6,2} 4) 2n2 – 4n – 14 = 0

  10. 1.7 Completing the Square: Solving with a≠1 Solve by completing the square. 1) 2x2 + 8x + 14 = 0 x2 + 4x + 7= 0 x2 + 4x + ___ = -7 + ___ (x + 2)2 = -3 x + 2 = Divide each term by 2 22 4

  11. Solve by Completing the Square 2) 2x2 – 5x + 3 = 0

  12. Standardized Test Practice SOLUTION Use the formula for the area of a rectangle to write an equation.

  13. 4) 2n2 – 4n – 14 = 0

  14. 1.7 Writing in vertex form • Write y = x2 + 2x + 4 in vertex form. Identify the vertex.

  15. Write in vertex form and identify the vertex 2.

  16. Write in vertex form. Identify the vertex. 3. y = -2x2 – 4x + 2

  17. Find the maximum value of a quadratic function Baseball The height y(in feet) of a baseball tseconds after it is hit is given by this function: y = –16t2 + 96t + 3 Find the maximum height of the baseball. SOLUTION The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

  18. ANSWER The vertex is (3, 147), so the maximum height of the baseball is 147feet. Find the maximum value of a quadratic function Write original function. y = – 16t2+ 96t +3 Factor –16 from first two terms. y = – 16(t2– 6t)+3 Prepare to complete the square. y +(–16)(?)= –16(t2 –6t + ?) + 3 y +(–16)(9)= –16(t2 –6t + 9) + 3 Add (-16)(9) to both sides y – 144 = –16(t – 3)2 + 3 Write t2 – 6t + 9 as a binomial squared. y = –16(t – 3)2+ 147 Solve for y.

  19. Warm-Up {0,9} • Solve by factoring: • Solve by completing the square: • Write in vertex form and identify the vertex.

  20. Warm-Up Solve by completing the square: • x2 + 4x – 12 = 0 • x2 + 2x + 3 = 0 • 3x2 – 2x + 1 = 0 {-6,2}

  21. 1.7 Examples Solve {-15,1} • x2 + 14x – 15 = 0 • x2 – 10x +13 = 0 • x2 + 4x – 12 = 0 • x2 + 2x + 3 = 0 • 3x2 – 2x + 1 = 0 {5 + 2 } {-6,2}

  22. Warm-Up • Solve by factoring: • x2 +9 = 6x • -2x2 + 12x – 16 = 0 • Solve by using square roots: • x2 + 10x +25 = 49 • Solve by completing the square. • 3x2 – 4x – 2 = 0

  23. Warm-Up • Solve by graphing: • Solve by factoring: • Solve by completing the square: {-3, 1} {0,9}

  24. Warm-Up • Solve by factoring: 1. 2x2 – 3x = -1 • Solve by completing the square: • x2 – 8x – 65=0 • 2x2 – 18x – 7=0 {1, ½} {-5,13}

  25. Warm-Up Solve. • 8x2 – 11 = 181 • -5(a+4)2 = 30 Solve by completing the square. 3) 3x2 – 2x + 1 = 0 Simplify: 4) 5) Write in vertex form 6)

  26. 1.8 Quadratic Formula and discriminant • Quadratic formula • Discriminant b2 -4ac • b2 – 4ac > 0 2 real roots (rational or irrational?) • b2 -4ac = 0 1 real root • b2 – 4ac < 0 2 imaginary roots

  27. 1.8 Examples Use the quadratic formula to solve. • x2 – 8x = 33 • x2 - 34x + 289 = 0 • x2 – 6x + 2 = 0 • x2 + 13 = 6x

  28. 1.8 Examples continued • Find the value of the discriminant and describe the nature and types of roots. • x2 + 6x + 9 = 0 • x2 + 3x + 5 = 0 • x2 + 8x – 4 =0 • x2 – 11x + 10 = 0

  29. h(t) = –16t2 + v0t +h0 An object is thrown upward from a height of 15 feet at an initial velocity of 35 feet per second. How long will it take for the object to hit the ground?

  30. EXAMPLE 5 Solve a vertical motion problem Juggling A juggler tosses a ball into the air. The ball leaves the juggler’s hand 4feet above the ground and has an initial vertical velocity of 40feet per second. The juggler catches the ball when it falls back to a height of 3feet. How long is the ball in the air? SOLUTION Because the ball is thrown, use the model h(t) = –16t2 + v0t +h0. To find how long the ball is in the air, solve for twhen h = 3.

  31. Transparency 6 Click the mouse button or press the Space Bar to display the answers.

  32. Transparency 6a

  33. Warm-Up • Determine the discriminant and describe the nature of the roots for: x2 – 2x – 17=0 • Use the quadratic formula to solve #1. • Write in vertex form. Identify the vertex. A. y = x2 – 10x +32B. y = -4x2 +16x – 11 • Solve by completing the square. x2 – 8x – 65=0

  34. Warm-Up • Determine the discriminant and describe the nature of the roots for: x2 – 2x – 17=0 • Use the quadratic formula to solve #1. • h(t) = –16t2 + v0t +h0 An object is thrown upward from a height of 15 feet at an initial velocity of 35 feet per second. How long will it take for the object to hit the ground?

  35. Warm-Up • Solve 5x2 – 5x + 7=3x2 + 10 using the quadratic formula. Write in vertex form and identify the vertex, axis of symmetry, and direction of the opening. • y=x2 – 8x + 9 • y=-2x2 – 8x – 1 {-1/2, 3} y=(x – 4)2 – 7, (4,-7), x=4, up y=-2(x + 2)2 + 7, (-2, 7), x=-2, down & skinny

  36. After the quiz … • For EXTRA CREDIT in your portfolio you may do the Practice Quiz 2 on p.328 #’s 1 – 10. This would be turned in as part of your unit portfolio.

  37. 1.9 Graphing and solving quadratic inequalities • Graph the inequality and decide if the parabola should be solid or dashed. • Test a point inside the parabola to see if it is a solution. • If it is a solution, shade inside. • If it is not a solution, shade outside. • Or if you use the Inequalities App on the TI-84 and it will do this for you!

  38. Warm-Up Lesson 1.9 • Graph: y > x2 – 3x + 2 • Graph the system of inequalities: y>x2 – 3 y < -2x2 + 4x + 2 Solve by graphing or algebraically • 0 < -2x2 – 6x + 1 • x2 – 4x + 3 > 0 • x2 + x < 2

  39. Warm-Up Graph: 1) y < x2 + 4x 2) y < -x2 – 3x + 10 Solve: 3) x2 + 3x – 28 < 0 4) x2 + 2x > 24 3. -7<x<4 4. x<-6 or x>4

  40. Warm-Up • Graph: y < -x2 – 3x + 10 Solve. 2) x2 – 3x – 18 > 0 3) x2 – 4x < 5 2. x<-3 or x>6 3. -1<x<5

  41. Chapter 1 Study Guide • Determine y-intercept, vertex, axis of symmetry and maximum or minimum values. • Solve by factoring • Solve by completing the square • Solve using quadratic formula • Solve using graphing calculator • Give the discriminant and describe the nature of the roots • Write a quadratic equation with the given roots. • Write equations in vertex form, analyze, and graph. • Graph and solve quadratic inequalities.

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