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Factoring a Trinomial and Completing the Square

Factoring a Trinomial and Completing the Square. Lesson 7.3. Let’s review multiplying (x + 3)(x + 4) we can use a multiplication rectangle to help us. x. + 4. x 2. 4x. x. 12. 3x. + 3. (x + 3)(x + 4) =x 2 + 3x + 4x + 12 = x 2 + 7x + 12.

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Factoring a Trinomial and Completing the Square

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  1. Factoring a Trinomial and Completing the Square Lesson 7.3

  2. Let’s review multiplying (x + 3)(x + 4) we can use a multiplication rectangle to help us. x + 4 x2 4x x 12 3x + 3 (x + 3)(x + 4) =x2 + 3x + 4x + 12 = x2 + 7x + 12

  3. Multiply (x - 2)(x - 5) we can use a multiplication rectangle to help us. x - 5 x2 -5x x + 10 -2x - 2 (x - 2)(x - 5) =x2 - 2x - 5x + 10 = x2 - 7x + 12

  4. Multiply (3x + 2)(2x - 5) we can use a multiplication rectangle to help us. 2x - 5 6x2 -15x 3x - 10 4x + 2 (3x + 2)(2x - 5) =6x2 + 4x - 15x - 10 = x2 - 11x - 10

  5. Try multiplying (4x + 1)(3x + 5) Try multiplying (3x - 1)(2x + 3) (4x + 1)(3x + 5) = 12x2 + 20x + 3x + 5 = 12x2 + 23x + 5 (3x - 1)(2x + 3) = 6x2 + 9x - 2x - 3 = 6x2 + 7x - 3

  6. Factoring is the reverse of multiplying. We can still use the multiplication rectangle to help us. Suppose we want to factor x2 + 8x + 15 x + 5 x2 5x x 15 3x You may find it helpful to consider all factors that make 15. + 3 x2 + 8x + 15= x2 + __x + ___x + 15 = (x + 5)(x + 3)

  7. Factor x2 + 2x - 15 x + 5 x2 5x x -15 - 3x You may find it helpful to consider all factors that make -- 15. Then pick out the pair that adds to -2 - 3 x2 + 2x - 15= x2 + __x + ___x - 15 = (x + 5)(x - 3)

  8. Factor 3x2 - 2x - 5 3x - 5 3x2 - 5x x - 5 3x You may find it helpful to consider all factors that make -- 15. Then pick out the pair that adds to -2 + 1 3x2 - 2x + 5= x2 + __x + ___x - 15 = (3x - 5)(x + 1)

  9. Factor 4x2 - 13x + 9 x - 1 4x2 - 4x 4x - 9x + 9 - 9 Which pair adds to -13? 4x2 - 13x + 8= x2 + __x + ___x + 8 = (4x - 9)(x - 1)

  10. Factor these trinomials using the multiplication rectangle and the sets of factors. The last three are called perfect squares.

  11. Complete the Square • Complete a rectangle diagram to find the product (x+5)(x+5), which can be written (x+5)2. • Write out the four-term polynomial, and then combine any like terms you see and express your answer as a trinomial. x2 5x x +5 5x 25 x +5 x2+5x+5x+25=x2+10x+25

  12. What binomial expression is being squared, and what is the perfect-square trinomial represented in the rectangle diagram at right? • Use a rectangle diagram to show the binomial factors for the perfect-square trinomial x2+24x+144.

  13. Find the perfect-square trinomial equivalent to (a+b)2 = . • Describe how you can find the first, second, and third terms of the perfect square trinomial (written in general form) when squaring a binomial. a + b a + b

  14. Not all polynomials are perfect square trinomials, but it is possible to complete the square on many of these other polynomials.

  15. Consider the expression x2+6x. • Where would the x2 fit? • Where would the 6x fit? • What would be placed in the last section to have a perfect square? • But we must carefully write this out algebraically. So

  16. Let’s compare the graphs of What is the vertex of the parabola? Which form shows the vertex in the equation? We call this new form the vertex form for a parabola. (x-h)2 + k

  17. Consider the expression x2 + 8x + 4. • Where would you place x2? • Where would you place the 8x? • What number would you like to have in the last section to have a perfect square? • Rewrite the expression x2 + 8x + 4 in the form (x-h)2 + k.Use a graph or table to verify that your expression is equivalent to the original expression, x2 + 8x + 4 . The vertex is located at (-4,-12)

  18. Rewrite each expression in the form (x-h)2+k. If you use a rectangle diagram, focus on the 2nd- and 1st-degree terms first. Verify that your expression is equivalent to the original expression.

  19. When the 2nd-degree term has a coefficient, you can first factor it out of the 2nd- and 1st-degree terms. For example, 3x2 +24x+5 can be written 3(x2 +8x)+5 . Completing a diagram for x2 +8x can help you rewrite the expression in the form a(x-h)2+k.

  20. Rewrite each expression in the form (x-h)2+k. • Use a graph or table to verify that your expression for (a) is equivalent to the original expression.

  21. Example B • Convert each quadratic function to vertex form. Identify the vertex.

  22. If you graph the quadratic function y=ax2+bx+c, what will be the x-coordinate of the vertex in terms of a, b, and c? • How can you use this value and the equation to find the y-coordinate?

  23. Example C • Nora hits a softball straight up at a speed of 120 ft/s. If her bat contacts the ball at a height of 3 ft above the ground, how high does the ball travel? When does the ball reach its maximum height? Using the projectile motion function, you know that the height of the object at time x is represented by the equation . The initial velocity, v0, is 120 ft/s, and the initial height, s0, is 3 ft. Because the distance is measured in feet, the approximate leading coefficient is 16. Thus, the function is y=-16x2 +120x + 3.

  24. To find the maximum height, locate the vertex. • The softball reaches a maximum height of 228 ft at 3.75 s.

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