Completing the Square

Slideshow 20 　Mathematics

Mr Sasaki　　　 Room 307

Objectives• Practice solving equations in the form • Solve quadratic equations by completing

the square where there is an even coefficient of

• Solve quadratic equations by completing the square where there is an odd coefficient of

Solving EquationsLet’s do a nice, easy warm up!ExampleSolve .

(𝑥+4 )2=25⇒𝑥+4=±5⇒𝑥=−9𝑜𝑟 1Very easy isn’t it!

Answers𝑥=−4𝑜𝑟 0 𝑥=4𝑜𝑟 6𝑥=−10𝑜𝑟 4 𝑥=7𝑥=−15𝑜𝑟 3 𝑥=−11𝑜𝑟 15

𝑥=− 112 𝑜𝑟52 𝑥=− 92 𝑜𝑟

112

𝑥=−5𝑜𝑟 −2 𝑥=−7 𝑜𝑟 12𝑥=√3−1 𝑥=√5+2𝑥=7√2−5 𝑥=3√3+6𝑥=√15+39

6 𝑥=√21−11718

Completing the SquareWe did this in Grade 8! We need to

transform one side of the equation to make a perfect square.ExampleWrite what you need to add to make a perfect square for…𝑥2+6 𝑥+¿9𝑥2−4 𝑥+¿4

𝑥2+𝑥+¿14We half and square the coefficient. How do they factorise?

¿¿¿

(𝑥+3 )2

(𝑥−2 )2

(𝑥+12 )

2

These values are half of the -coefficients.

Completing the SquareWe can use this concept to write some quadratic equation in the form where .ExampleBy completing the square, solve .𝑥2−10 𝑥+21=0⇒𝑥2−10 𝑥=−21Note: We must write it in the form .

⇒𝑥2−10𝑥+¿−21+¿25 25( 12 ∙10)

2

⇒ (𝑥−5 )2=4⇒𝑥−5=±2

or

Completing the SquareLet’s try another example.ExampleBy completing the square, solve .2 𝑥2−12𝑥+8=0⇒𝑥2−6 𝑥+4=0

⇒𝑥2−6 𝑥=−4⇒𝑥2−6 𝑥+¿−4+¿9 9

⇒ (𝑥−3 )2=5⇒𝑥−3=√5⇒𝑥=√5+3

Answers16 4 1 149 7 196 14

or or

𝑥=√6−3 or or 𝑥=2√3+4

Completing the SquareWorking with odd numbers makes things messier.ExampleBy completing the square, solve .𝑥2+𝑥−2=0⇒𝑥2+𝑥=2

⇒𝑥2+𝑥+¿2+¿14

⇒(𝑥+12 )

2

=94

⇒𝑥+12=±

32

or

14

Answers - Easy2.25 1.5 6.25 2.5

20.25 4.5 56.25 7.5

or or

𝑥=1or   4 or or 𝑥=−7  or  18

9432

254

52

814

92

2254

152

Answers - Hard2 √2 √3

234

3√545 √24

18

or or

or or (both negative roots)

or (positive) or (positive)