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Complete the square - Higher

Complete the square - Higher

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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


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


PROCEDURE

1.1.​​

Write the quadratic expression in descending powers of xx​. 

2.2.​​

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

33.

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

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.​​

Add the constant term.


Example 1

Complete the square on the quadratic expression

x2+6xx^2+6x​​


Start by setting up the answer in the form

(x)2(x \qquad)^2


Take the co-efficient of xx, in this case 66and half the number, which gives 33​. Fill this number in the bracket.

(x+3)2(x+3)^2​​


Take the number in the bracket, 33, and square it. Subtract this number from the bracket. This gives​

(x+3)29\underline{(x+3)^2-9}


Example 2

Complete the square on the quadratic expression

x2+10x+6x^2+10x+6​​


Start by setting up the answer in the form

(x)2(x \qquad)^2


Take the co-efficient of xx, in this case 1010 and half the number, which gives 55​. Fill this number in the bracket.

(x+5)2(x+5)^2​​


Take the number in the bracket, 55, and square it. Subtract this number from the bracket and add the constant term. This gives

(x+5)225+6(x+5)219(x+5)^2-25+6 \\ \underline{(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=0​​


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

(x+5)219=0(x+5)^2-19=0


To solve the quadratic, take 1919 to the other side of the equation and then square root, before taking 55 over.

(x+5)219=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}​​


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

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


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Exercises

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FAQs - Frequently Asked Questions

What are the different ways of solving a quadratic?

How do you complete the square?

What is 'completing the square'?

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