Completing the Square: Comprehensive Guide With Worksheet

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Introduction

  • Completing the squares is a powerful technique used in algebra to solve quadratic equations.
  • It is important to study the Completing the Square because it helps solve quadratic equations, understand the shape of parabolas, find the turning point etc.

What is Completing the Square?

  • Completing the square is an algebraic technique that transforms a quadratic expression into the sum of a perfect square binomial and a remaining constant term.

Diagram showing how to rearrange a quadratic expression into completed square form, transforming ax² + bx + c into a(x + d)² + e for GCSE Completing the Square.

  • This method reveals key properties of quadratic functions.

Steps to Solve Completing the Square

  • Completing the square is a method used to rewrite quadratic expressions in the form:

Formula showing the completed square or vertex form of a quadratic, a(x - h)² + k, used in Completing the Square for GCSE Maths.

Steps to convert Quadratic expression into the form of a(x – h) + k by Completing the Square:

  • Step #1: Identify the Quadratic expression

Standard form of a quadratic expression ax² + bx + c, used as the starting point for Completing the Square in GCSE Maths.

  • Step #2: Solve for:

Expression b divided by 2a, representing the step used in Completing the Square when working from ax² + bx + c.

  • Step #3: Re-write the equation as,

Expression showing a(x + b ÷ 2a)² + k, representing the completed square form of a quadratic in GCSE Completing the Square.

  • Step #4: Plug x = -b/2a to your original equation and solve for k.
  • Step #5: Simplify it.

We can also find the Turning point by Completing the Square

  • After finding the equation:

Expression showing a(x + b ÷ 2a)² + k, representing the completed square form of a quadratic in GCSE Completing the Square.

Identify the Turning Point:

Comparing with,

Vertex form of a quadratic y = a(x − h)² + k, used in Completing the Square to identify the turning point of a parabola.

where,

  • h and k are the turning point (vertex) of the parabola.
  • a determines the parabola’s width and direction (upwards if a>0, downwards if a<0).

certified Physics and Maths tutorSolved Example:

Problem: Factorize 2x2+6x+9 by completing the square.

Solution: 

Step #1: Identify the Quadratic expression.

Here,

    • a = 2 (coefficient of x2)
    • b = 6 (coefficient of x)
    • c = 9 (constant term)

Step #2: Solve for:

Example of b divided by 2a in Completing the Square, showing 6 ÷ 2(2) = 3/2 for a quadratic expression.

Step #3: Re-write the equation as:

Example of the completed square form using values a=2 and b÷2a=3/2, showing 2(x + 3/2)² + k in the Completing the Square method for GCSE Maths.

Step #4: Plug x = -b/2a to your original equation and solve for k.

Worked example expanding 2(x + 3/2)² during Completing the Square, showing intermediate steps including squaring -3/2 and multiplying to form 18/2 and 9/2.

Step #5: Simplify it.

Final simplified expression after expanding the completed square bracket: 2x² + 3x + 9/2, showing expansion result in Completing the Square.

certified Physics and Maths tutorSolved Example:

Problem: Find the turning point of y = 2(x−2)2− 3

Solution: 

Step #1: Identify the Turning Point:

Example of vertex form after Completing the Square: y = 2(x − 2)² + 3, showing quadratic in completed square format for GCSE students.

    • h = 2 (the x-coordinate of the vertex).
    • k = −3 (the y-coordinate of the vertex).

The Turning point (vertex) of the parabola is (2, -3).

certified Physics and Maths tutorSolved Example:

Problem: Factorize x2+6x+5 by completing the square.

Solution: 

Step #1: Identify the Quadratic expression.

Here,

    • a = 1 (coefficient of x2)
    • b = 6 (coefficient of x)
    • c = 5 (constant term)

Step #2: Solve for:

Example calculating b ÷ 2a in Completing the Square, showing 6 ÷ 2(1) = 3.

Step #3: Re-write the equation as:

Completed square form a(x + b/2a)² + k shown numerically as x + 3 squared plus k for the Completing the Square method.

Step #4: Plug x = -b/2a to your original equation and solve for k.

Worked example expanding and simplifying −3² and 6×−3 plus 5 during Completing the Square, showing 9, 18, 5, and −4.

Step #5: Simplify it.

Final completed square form of a quadratic shown as (x − 3)² − 4, representing the result of Completing the Square.

certified Physics and Maths tutorSolved Example:

Problem: Find the turning point of 

Example of a quadratic written in vertex form after Completing the Square: y = 3(x + 5/3)² + 4/3.

Solution: 

Step #1: Identify the Turning Point:

Example of a quadratic written in vertex form after Completing the Square: y = 3(x + 5/3)² + 4/3.

    • h = -5/3 (the x-coordinate of the vertex).
    • k = −4/3 (the y-coordinate of the vertex).

The Turning point (vertex) of the parabola is (-5/3, -4/3).

certified Physics and Maths tutorSolved Example:

Problem: Given the quadratic equation:

Quadratic equation y = 2x² + 8x + 5 as the starting form before Completing the Square in GCSE Maths.

Rewrite the equation in completed square form and hence, find the turning point of the parabola.

Solution: 

Step #1: Identify the Quadratic expression.

Here,

    • a = 2 (coefficient of x2)
    • b = 8 (coefficient of x)
    • c = 5 (constant term)

Step #2: Solve for:

Example of the Completing the Square step b ÷ 2a using 8 ÷ 2(2) = 2.

Step #3: Re-write the equation as:

Completed square structure a(x + b/2a)² + k shown numerically as (x + 2)² + k when Completing the Square.

Step #4: Plug x = -b/2a to your original equation and solve for k.

Worked example expanding 2(x − 2)² and simplifying values including squaring −2 to get 4, forming 8, 16, 5, and −3 during Completing the Square.

Step #5: Simplify it.

Final completed square form written as y = 2(x − 2)² − 3, representing the result after Completing the Square.

Step #6: Identify the Turning Point:

    • Vertex form:

Formula showing the completed square or vertex form of a quadratic, a(x - h)² + k, used in Completing the Square for GCSE Maths.

Comparing with it,

Turning point values from Completing the Square: h = −2 and k = −3, identifying the vertex of the quadratic graph.

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