Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations Perfect Square Trinomials and Completing the Square Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Completing the Square A quadratic relation in standard form can be rewritten in vertex form by creating a perfect square in the expression, then factoring the square. This technique is called completing the square. Completing the square can be used to find the vertex of a quadratic in standard form without finding the zeros of the relation or two points equidistant from the axis of summetry. Completing the square allows you to find the maximum or minimum value of a quadratic relation algebraically, without using a graph. Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

Completing the Square (Steps) Remove the common constant factor from both the x2 and x term. Find the constant that must be added and subtracted to create a perfect square. This value equals the square of half of the coefficient of the x –term in step 1. Rewrite the expression by adding, then subtracting, this value after the x – term inside the brackets. Group the three terms that form the perfect square. Move the subtracted value outside the brackets by multiplying it by the common constant factor. 4) Factor the perfect square and collect like terms. Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

+16 - 16 +9 - 9 Example 1: Completing the Square Convert each of the following into vertex form by completing the square. Think about: +16 - 16 Think about: +9 - 9 Perfect Trinomial Perfect Trinomial Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

25 + - 4 Example 1: Completing the Square - Continued Convert each of the following into vertex form by completing the square. 25 4 + - Think about: Perfect Trinomial Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

+4 -4 Example 2: Convert to Vertex Form by Completing the Square Convert each of the following into vertex form by completing the square. Determine the vertex, axis of symmetry, opening, intercepts and the mapping rule. Common factor for first two terms only! +4 -4 Think about: To create Perfect Trinomial, multiply the last term -4 to common factor -1! Perfect Trinomial Last term in standard form Mapping Rule: Vertex: Parabola opens (concaves) down, translates 2 units left and up 9 units. Opens down Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

+4 -4 Example 2: Convert to Vertex Form by Completing the Square Convert each of the following into vertex form by completing the square. Determine the vertex, axis of symmetry, opening, intercepts and the mapping rule. Common factor for first two terms only! +4 -4 Think about: To create Perfect Trinomial, multiply the last term -4 to common factor 3! Perfect Trinomial Vertex: Mapping Rule: Last term in standard form Parabola opens (concaves) up, vertical expands by factor of 3, translates 2 units left and down 17 units. Opens up Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

25 + - 16 Example 2: Convert to Vertex Form by Completing the Square Convert each of the following into vertex form by completing the square. Determine the vertex, axis of symmetry, opening, intercepts and the mapping rule. Common factor for first two terms only! 25 16 + - Think about: To create Perfect Trinomial, multiply the last term -25/16 to common factor 2! Perfect Trinomial Mapping Rule: Vertex: Last term in standard form Parabola opens (concaves) up, vertical expands by factor of 2, translates 1.25 units left and down 33/8 units. Opens up Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

Quadratic formula! - Roots (x intercepts) - Vertex - y intercept TRY: Convert to Vertex Form by Completing the Square Convert each of the following into vertex form by completing the square. Determine the vertex, axis of symmetry, opening, intercepts and the mapping rule. Quadratic formula! - Roots (x intercepts) - Vertex - y intercept Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Perfect Square Trinomials and Completing the Squares Text: P. 390 #1, 2, 4, 5, 6, 8, 9 Check the website for updates Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Perfect Square Trinomials and Completing the Square © 2017 E. Choi – MPM2D - All Rights Reserved