2. OCF.01.3 - Completing the Square Technique MCR3U - Santowski

3. (A) Perfect Square Trinomials Polynomials like (x + 4) 2 are called perfect squares. Polynomials like (x + 4) 2 are called perfect squares. When expanded, (x + 4) 2 equals x 2 + 8x + 16. When expanded, (x + 4) 2 equals x 2 + 8x + 16. When x 2 + 8x + 16 is factored, it becomes (x + 4) 2 When x 2 + 8x + 16 is factored, it becomes (x + 4) 2 Notice the relationship between the x term (8) and the constant term (16) ????? Notice the relationship between the x term (8) and the constant term (16) ????? If not, try another perfect square, say (x + 6) 2 and x 2 + 12x + 36 (12 and 36??) If not, try another perfect square, say (x + 6) 2 and x 2 + 12x + 36 (12 and 36??) ex. Find the value of b that makes each expression a perfect square ex. Find the value of b that makes each expression a perfect square (a) x 2 + 4x + b (a) x 2 + 4x + b (b) x 2 + 10x + b (b) x 2 + 10x + b (c) x 2 + 14x + b (c) x 2 + 14x + b (d) x 2 - 12x + b (d) x 2 - 12x + b (e) x 2 – 5x + b (e) x 2 – 5x + b

4. (B) The Vertex Form of a Quadratic Equation The quadratic equation y = a(x – h) 2 + k is the vertex form and conveniently gives us the location of the vertex of the parabola, which is at (h,k). The quadratic equation y = a(x – h) 2 + k is the vertex form and conveniently gives us the location of the vertex of the parabola, which is at (h,k). In many cases the equation of a quadratic may be presented in standard form y = ax 2 + bx + c In many cases the equation of a quadratic may be presented in standard form y = ax 2 + bx + c We have to go through the “completing the square” technique in order to convert the equation to the vertex form. We have to go through the “completing the square” technique in order to convert the equation to the vertex form. Then, we can identify the vertex and if the vertex represents a maximum (parabola opens down) or a minimum (parabola opens up) by the sign of the a term. If a is positive, the parabola opens up and the vertex represents a min. point and if a is negative, the parabola opens down and the vertex represents a max. point. Then, we can identify the vertex and if the vertex represents a maximum (parabola opens down) or a minimum (parabola opens up) by the sign of the a term. If a is positive, the parabola opens up and the vertex represents a min. point and if a is negative, the parabola opens down and the vertex represents a max. point.

5. (C) The Completing the Square Method Steps Steps Involved in the technique: Example Example using y = 2x 2 2x 2 + 12x - 3 (i) Factor (i) Factor the leading co-efficient of 2 (the value of a) a) from both the x2 x2 x2 x2 and x terms y = 2(x 2 2(x 2 + 6x) - 3 (ii) (ii) Find the constant term that must be present in order to create a perfect square trinomial y = 2(x 2 2(x 2 + 6x +9 ) - 3 BUT *****????? (iii) (iii) Make an adjustment in the equation because of the extra value added (+9) y = 2(x 2 2(x 2 + 6x + 9 - 9) - 3 (iv) (iv) Group the perfect square trinomial. Move the subtracted value out of the bracket y = 2(x 2 2(x 2 + 6x + 9) – 2x9 - 3 (v) (v) Factor the perfect square and collect like terms y = 2(x + 3) 2 3) 2 - 21

6. (D) Further Examples of the Method ex 1. y = x 2 + 8x + 15 ex 1. y = x 2 + 8x + 15 ex 2. y = -2x 2 + 12x - 7 ex 2. y = -2x 2 + 12x - 7 ex 3. y = -0.3f 2 - 2.4f + 7.3 ex 3. y = -0.3f 2 - 2.4f + 7.3 ex 4. y = 5x – 3x 2 ex 4. y = 5x – 3x 2 In each example, identify the vertex and the direction of opening of the parabola In each example, identify the vertex and the direction of opening of the parabola

7. Internet Links Completing the Square by James Brennan Completing the Square by James Brennan Completing the Square by James Brennan Completing the Square by James Brennan Completing the Square: Solving Quadratics from Purple Math Completing the Square: Solving Quadratics from Purple Math Completing the Square: Solving Quadratics from Purple Math Completing the Square: Solving Quadratics from Purple Math Completing the Square from Bethany Lutheran College Completing the Square from Bethany Lutheran College Completing the Square from Bethany Lutheran College Completing the Square from Bethany Lutheran College

8. (E) Homework Nelson Text, p306, Q1abcdefg, 2abcdef, 3abc, 5abcd, 7, 9 Nelson Text, p306, Q1abcdefg, 2abcdef, 3abc, 5abcd, 7, 9