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Complete the square - Higher

Complete the square - Higher

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Explainer Video

Tutor: Bilal

Summary

Complete the square

​​In a nutshell

Completing the square is a method that can be used to solve quadratic equations. It involves rearranging the quadratic into a form which can make it easier to solve. It is another way of solving a quadratic equation, instead of factorising or using the quadratic formula.



Complete the square

​To complete the square on a quadratic expression, it should be put into the form

​(x+a)2+b(x+a)^2+b(x+a)2+b


Once the quadratic has been put in the completed square form, it can be rearranged to solve for xxx.


PROCEDURE

​1.1.1.​​

Write the quadratic expression in descending powers of xxx​. 

​2.2.2.​​

Set up the answer in the form (x)2(x \qquad )^2(x)2.​

​333.

Take the co-efficient of xxx from the quadratic, half the number and fill this number in the bracket.​

​4.4.4.​​

Take the number that has just been filled in the bracket, square the number and subtract this number from the bracket.

​5.5.5.​​

Add the constant term.


Example 1

Complete the square on the quadratic expression

​x2+6xx^2+6xx2+6x​​


Start by setting up the answer in the form

​(x)2(x \qquad)^2(x)2


Take the co-efficient of xxx, in this case 666​ and half the number, which gives 333​. Fill this number in the bracket.

(x+3)2(x+3)^2(x+3)2​​


Take the number in the bracket, 333, and square it. Subtract this number from the bracket. This gives​

​(x+3)2−9‾\underline{(x+3)^2-9}(x+3)2−9​


Example 2

Complete the square on the quadratic expression

​x2+10x+6x^2+10x+6x2+10x+6​​


Start by setting up the answer in the form

​(x)2(x \qquad)^2(x)2


Take the co-efficient of xxx, in this case 101010​ and half the number, which gives 555​. Fill this number in the bracket.

(x+5)2(x+5)^2(x+5)2​​


Take the number in the bracket, 555, and square it. Subtract this number from the bracket and add the constant term. This gives

​(x+5)2−25+6(x+5)2−19‾(x+5)^2-25+6 \\ \underline{(x+5)^2 - 19}(x+5)2−25+6(x+5)2−19​



Solve a quadratic by completing the square

Once a quadratic has been rearranged into the completed square form, it is possible to solve the equation.


Example 3

Solve

​x2+10x+6=0x^2+10x+6=0x2+10x+6=0​​


This has already been rearranged into the correct form in example 2 above.

​(x+5)2−19=0(x+5)^2-19=0(x+5)2−19=0


To solve the quadratic, take 191919 to the other side of the equation and then square root, before taking 555 over.

(x+5)2−19=0(x+5)2=19x+5=±19x=−5±19\begin {aligned}(x+5)^2-19&=0 \\(x+5)^2 &= 19 \\x+5 &= \pm \sqrt {19} \\x &= -5 \pm \sqrt {19}\end {aligned}(x+5)2−19(x+5)2x+5x​=0=19=±19​=−5±19​​​​


The answer can be left in surd form, or can be calculated to 3.s.f.3.s.f.3.s.f.​​

​x=−9.36 or x=−0.641‾\underline{x= -9.36 \space or \space x=-0.641}x=−9.36 or x=−0.641​​​


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Factorising quadratics

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Factorising quadratics

The quadratic formula - Higher

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Complete the square - Higher

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FAQs - Frequently Asked Questions

What are the different ways of solving a quadratic?

You can solve a quadratic by factorising, using the quadratic formula or completing the square.

How do you complete the square?

You can complete the square by rearranging a quadratic expression into the form (x+a)^2+b.

What is 'completing the square'?

Completing the square is a method that can be used to solve quadratic equations.

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