Solving Quadratic Identities

March 12, 2023

Differentiated Learning Objectives

• All students should solve a quadratic identity in the form (x + b)² + c by completing a square.
• Most students should solve a quadratic identity in the form (x + b)² + c by completing a square and equating terms.
• Some students should solve a quadratic identity in the form a(x + b)² + c.

Links to Lesson Resources (Members Only)

Starter/Introduction

Encourage students to work in pairs and discuss the minimum possible value a squared number or term could have. Next, students should write the expressions in ascending order of their minimum possible value. Understanding this will later help students when finding the turning point in sketched graphs.

 Prompts / Questions to consider

• Can a squared number or term ever result in a negative?
• What is the smallest possible result when any number has been squared?

Solving Quadratic Identities

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A quadratic identity can be solved either by expansion and equating terms or by completing the square. Students should be taught both ways and encouraged to use one method to check the accuracy of the other.

 Prompts / Questions to consider

• How can we use the method of completing the square to make the left-hand side of the identity equal to the right-side?
• How can we equate terms to make the right-hand side of the identity equal to the left-hand side?

Plenary

The plenary challenges students to solve a variety of quadratic identities across the ability range. Encourage more able students to focus on the quadratics 9x² – 12 + 9 and 4x² + 12x + 4. This activity takes between 5 to 10 minutes for a higher-ability class.

 Prompts / Questions to consider

• Which expressions are more suited to equating terms or completing the square?
• Is there a relationship between the constant term in the brackets of the completed square form and the b term in the expanded quadratic?

Differentiation

When solving quadratic identities more able students could solve identities in the form ax² + bx + c.  Making the coefficient of x an odd number also increases the level of difficulty. Less able students may benefit from equating terms of quadratics in the form x² + bx + c where b is even and positive.