Warm Up Lesson Presentation Lesson Quiz · PDF file Holt Algebra 1 Warm Up Lesson Presentation...

Click here to load reader

  • date post

    26-Sep-2020
  • Category

    Documents

  • view

    5
  • download

    0

Embed Size (px)

Transcript of Warm Up Lesson Presentation Lesson Quiz · PDF file Holt Algebra 1 Warm Up Lesson Presentation...

  • Holt McDougal Algebra 1

    8-8 Completing the Square8-8 Completing the Square

    Holt Algebra 1

    Warm Up

    Lesson Presentation

    Lesson Quiz

    Holt McDougal Algebra 1

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Warm Up

    Simplify.

    19 1. 2.

    3. 4.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Warm Up

    Solve each quadratic equation by factoring.

    5. x2 + 8x + 16 = 0

    6. x2 – 22x + 121 = 0

    7. x2 – 12x + 36 = 0

    x = –4

    x = 11

    x = 6

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Solve quadratic equations by completing the square.

    Objective

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    completing the square

    Vocabulary

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square.

    When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.

    X2 + 6x + 9 x2 – 8x + 16 Divide the coefficient of the x-term by 2, then

    square the result to get

    the constant term.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 1: Completing the Square

    Complete the square to form a perfect square trinomial.

    A. x2 + 2x + B. x2 – 6x +

    x2 + 2x

    x2 + 2x + 1

    x2 + –6x

    x2 – 6x + 9

    Identify b.

    .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 1

    Complete the square to form a perfect square trinomial.

    a. x2 + 12x + b. x2 – 5x +

    x2 + 12x

    x2 + 12x + 36

    x2 + –5xIdentify b.

    x2 – 5x +

    .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 1

    Complete the square to form a perfect square trinomial.

    c. 8x + x2 +

    x2 + 8x

    x2 + 8x + 16

    Identify b.

    .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Solving a Quadratic Equation by Completing the Square

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 2A: Solving x2 +bx = c by Completing the Square

    Solve by completing the square. Check your answer.

    x2 + 16x = –15

    Step 1 x2 + 16x = –15

    Step 2

    Step 3 x2 + 16x + 64 = –15 + 64

    Step 4 (x + 8)2 = 49

    Step 5 x + 8 = ± 7

    Step 6 x + 8 = 7 or x + 8 = –7

    x = –1 or x = –15

    The equation is in the

    form x2 + bx = c.

    Complete the square.

    Factor and simplify.

    Take the square root

    of both sides.

    Write and solve two

    equations.

    .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 2A Continued

    Solve by completing the square. Check your answer.

    x2 + 16x = –15

    The solutions are –1 and –15.

    Check x2 + 16x = –15

    (–1)2 + 16(–1) –15

    1 – 16 –15

    –15 –15✓

    x2 + 16x = –15

    (–15)2 + 16(–15) –15

    225 – 240 –15

    –15 –15✓

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 2B: Solving x2 +bx = c

    Solve by completing the square. Check your answer.

    x2 – 4x – 6 = 0

    Step 1 x2 + (–4x) = 6

    Step 3 x2 – 4x + 4 = 6 + 4

    Step 4 (x – 2)2 = 10

    Step 5 x – 2 = ± √10

    Write in the form

    x2 + bx = c.

    Complete the square.

    Factor and simplify.

    Take the square root

    of both sides.

    Write and solve two

    equations.

    Step 6 x – 2 = √10 or x – 2 = –√10

    x = 2 + √10 or x = 2 – √10

    .Step 2

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 2B Continued

    Solve by completing the square. Check your answer.

    The solutions are 2 + √10 and x = 2 – √10.

    Check Use a graphing calculator to check your answer.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 2a

    Solve by completing the square. Check your answer.

    x2 + 10x = –9

    Step 1 x2 + 10x = –9

    Step 3 x2 + 10x + 25 = –9 + 25 Complete the square.

    The equation is in the

    form x2 + bx = c.

    Step 2

    Step 4 (x + 5)2 = 16

    Step 5 x + 5 = ± 4

    Step 6 x + 5 = 4 or x + 5 = –4

    x = –1 or x = –9

    Factor and simplify.

    Take the square root

    of both sides.

    Write and solve two

    equations.

    .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 2a Continued

    Solve by completing the square. Check your answer.

    x2 + 10x = –9

    The solutions are –9 and –1.

    x2 + 10x = –9

    (–9)2 + 10(–9) –9

    81 – 90 –9

    –9 –9 ✓

    x2 + 10x = –9

    (–1)2 + 10(–1) –9

    1 – 10 –9

    –9 –9 ✓

    Check

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 2b

    Solve by completing the square. Check your answer.

    t2 – 8t – 5 = 0

    Step 1 t2 + (–8t) = 5

    Step 3 t2 – 8t + 16 = 5 + 16 Complete the square.

    Write in the form

    x2 + bx = c.

    Step 2

    Step 4 (t – 4)2 = 21

    Step 5 t – 4 = ± √21

    Factor and simplify.

    Take the square root

    of both sides.

    Write and solve two

    equations.

    Step 6 t = 4 + √21 or t = 4 – √21

    .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 2b Continued

    Solve by completing the square. Check your answer.

    t = 4 – √21 or t = 4 + √21.The solutions are

    Check Use a graphing calculator to check your answer.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 3A: Solving ax2 + bx = c by Completing the Square

    Solve by completing the square.

    –3x2 + 12x – 15 = 0

    Step 1

    x2 – 4x + 5 = 0 x2 – 4x = –5

    x2 + (–4x) = –5

    Step 3 x2 – 4x + 4 = –5 + 4

    Divide by – 3 to make a = 1.

    Write in the form x2 + bx = c.

    Complete the square.

    .Step 2

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 3A Continued

    Solve by completing the square.

    –3x2 + 12x – 15 = 0

    Step 4 (x – 2)2 = –1

    There is no real number whose square is negative, so there are no real solutions.

    Factor and simplify.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 3B: Solving ax2 + bx = c by Completing the Square

    Solve by completing the square.

    5x2 + 19x = 4

    Step 1 Divide by 5 to make a = 1.

    Write in the form x2 + bx = c.

    Step 2 .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Complete the square.Step 3

    Example 3B Continued

    Solve by completing the square.

    Factor and simplify.Step 4

    Step 5 Take the square root

    of both sides.

    Rewrite using like

    denominators.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Example 3B Continued

    Solve by completing the square.

    Step 6

    The solutions are and –4.

    Write and solve

    two equations.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 3a

    Solve by completing the square.

    3x2 – 5x – 2 = 0

    Step 1 Divide by 3 to make a = 1.

    Write in the form x2 + bx = c.

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Complete the square.

    Factor and simplify.

    Step 3

    Step 4

    Check It Out! Example 3a Continued

    Solve by completing the square.

    Step 2 .

  • Holt McDougal Algebra 1

    8-8 Completing the Square

    Check It Out! Example 3a Continued

    Solve by completing the square.

    Write and solve two

    equations.

    Step 6

    Take the square root

    of both sides.

    Step 5

  • Holt McDougal Algebra 1

    8-8 Com