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Completing The Square

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Completing The Square – Introduction

Introduction:
 Completing the square is a technique used in GCSE Higher Maths to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored or solved.

Understanding Quadratic Equations:
 To comprehend completing the square, it’s essential to understand quadratic equations. Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants.

The Process of Completing the Square:
 Step one: Rearrange the equation in the form ax^2 + bx = -c.
 Step two: Add (b/2)^2 to both sides of the equation.
 Step three: Rewrite the left side of the equation as a perfect square trinomial.
 Step four: Factor the perfect square trinomial.
 Step five: Take the square root of both sides of the equation.
 Step six: Solve for x by isolating it on one side.

Benefits of Completing the Square:
 Completing the square allows us to solve quadratic equations that are not easily factorable. It provides a reliable method to find the solutions, even when factoring is not feasible or straightforward.

Completing the Square - Examples

\begin{aligned} & (x+4)^2-16+6 \\ & (x+4)^2-10 \end{aligned}

\begin{aligned} & (x+6)^2-36-1 \\ & (x+6)^2-37 \end{aligned}

\begin{aligned} & (x-3)^2-9-3 \\ & (x-3)^2-12 \end{aligned}

\begin{gathered} (x-3)^2-12=0 \\ (x-3)^2=12 \\ x-3= \pm \sqrt{12} \\ x=3 \pm \sqrt{12} \end{gathered}

Completing the Square - Other Examples

\begin{aligned} & (x-3)^2-9+2=0 \\ & (x-3)^2-7=0 \\ & (x-3)^2=7 \\ & x-3= \pm \sqrt{7} \\ & x=3 \pm \sqrt{7} \end{aligned}

\begin{aligned} & (x-5)^2+31 \\ & (x-5)(x-5)+31 \\ & x^2-10 x+25+31 \\ & x^2-10 x+56 \end{aligned}
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Completing the Square – Use Of Fractions And Turning Points 


\begin{aligned} & \left(x-\frac{3}{2}\right)^2-\frac{9}{4}+7 \\ & \left(x-\frac{3}{2}\right)^2-\frac{9}{4}+\frac{28}{4} \\ & \left(x-\frac{3}{2}\right)^2+\frac{19}{4} \end{aligned}
 
 
\begin{gathered} 3\left(x^2+6 x-\frac{1}{3}\right) \\ 3\left[(x+3)^2-9-\frac{1}{3}\right] \\ 3\left[(x+3)^2-\frac{27}{3} \cdot \frac{1}{3}\right] \\ 3\left[(x+3)^2-\frac{28}{3}\right] \\ 3(x+3)^2-28 \end{gathered}
 
\begin{aligned} &\begin{aligned} & y=(x-3)^2-9+1 \\ & y=(x-3)^2-8 \end{aligned}\\ &(3,-8) \end{aligned}
 
\begin{aligned} & y=(x+2)^2-4+7 \\ & y=(x+2)^2+3 \\ & (-2,3) \end{aligned}

Conclusion:
 Completing the square is a powerful technique used to solve quadratic equations in GCSE Higher Maths. By manipulating the equation to create a perfect square trinomial, we can find the solutions more easily. This method is particularly useful when factoring is challenging or not possible.

If you decide to pursue A Level Maths you will find that you will revisit the topic of completing the square in the first half term of year 12. 

The process is revisited as it is used in a number of areas such as curve sketching, calculus and also in the area of mechanics. 

Once you have completed the square you are able to obtain important information about the shape of a quadratic curve such as its line of symmetry and also its vertex which is also known as its maximum or minimum turning point. 

Probably the hardest part of completing the square is having to take out a factor and also to be able to complete the process when having to use fractions. 

If you, or your parents would like to find out more, please just get in touch via email at info@exam.tips or call us on 0800 689 1272. 

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