Revision Quadratic Equation

Solving Quadratic Equation by Complete the Square Method.

By I Porter

Introduction

Factoring to solve quadratic equations is generally quite fast and easy, but many quadratics cannot be solved by factorisation.

For example, the quadratic x2 + 5x - 1 = 0, appears to be simple, but cannot be factorised. For these, we must use the COMPLETE THE SQUARE method or the Quadratic Formula.

The Complete the Square method is based on the following two expansion:

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

What is really important, is the relationship between ±2b and b2.

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Note : 2b ÷ 2( )2

= b2

This is an important step in the complete the square method of solving quadratic equations.

Simple Example 1: Solve the following quadratic equations by complete the square method.

x2 + 8x = 0 For ax2 + 2bx + b2 = 0 2b = (+8)

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hence, 2b

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

=+8

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

= +16To complete the square we need to add b2 = +16 to both sides.

x2 + 8x + 16 = 0 + 16

x2 + 8x + 16 = 16 Factorise LHS, simplify RHS.

(x + 4)2 = 16 Take √ of both sides.

x + 4 = ±4 Solve for x.

x = -4±4

i.e. x = -4 - 4 or x = -4 + 4

x = -8 or x = 0

The solution is x = -8 and x = 0.

This could have been solved faster by factoring the original quadratic equation.

x2 + 8x = 0

x (x + 8) = 0 therefore

x = 0 or x + 8 = 0

x = 0 or x = -8

We obtain the same result, but a lot faster!

Simple Example 2 x2 + 8x + 12 = 0

For ax2 + 2bx + b2 = 0 2b = (+8)

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hence, 2b

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

=+8

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

= +16To complete the square we need to add b2 = +16 to both sides.

x2 + 8x + 16 = -12 + 16

x2 + 8x + 16 = 4 Factorise LHS, simplify RHS.

(x + 4)2 = 4 Take √ of both sides.

x + 4 = ±2 Solve for x.

x = -4±2

x = -6 or x = -2

x2 + 8x = -12

Rearrange to x2 + bx = c

i.e. x = -4 - 2 or x = -4 + 2

The solution is x = -8 and x = 0.

Example 3x2 - 5x - 3 = 0 Rearrange to x2 + bx = c

x2 - 5x = 3 For ax2 - 2bx + b2 = 0 2b = (-5)

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hence, 2b

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

=−5

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

= +25

4

To complete the square we need to add b2 = +25/4 to both sides.

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x 2 − 5x +254

= 3+254

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x 2 − 5x +254

=374

Factorise LHS, simplify RHS.

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x −52

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

=374

Take √ of both sides.

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x −52

=± 37

2Solve for x.

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x =52

± 372

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x =5 − 37

2 or x =

5 + 372

The solutions should be written in exact form as above.

Exercise 1 Solve the following quadratic equations by complete the square method.

a) x2 + 6x = 0

b) x2 - 7x = 0

c) x2 - 10x = -9

d) x2 - 6x + 3 = 0

e) x2 + 9x - 5 = 0

f) x2 - 3x - 7 = 0

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x = 0 or x = −6

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x = 0 or x = 7

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x =1 or x = 9

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x = 3− 6 or x = 3+ 6

€

x =−9 − 101

2 or x =

−9 + 1012

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x =3− 37

2 or x =

3+ 372

Harder Examples: Solve by complete the square method.

a)

€

2x2 + 8x − 6 = 0 Divide by 2 and rearrange to x2 + bx = c

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x2 + 4x = 3 For ax2 + 2bx + b2 = 0 2b = (+4)

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hence, +4

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

= +4

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x2 + 4x + 4 = 3+ 4

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x2 + 4x + 4 = 7 Factorise LHS.

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x + 2( )2

= 7 Take √ of both sides.

Solve for x.

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x + 2 = ± 7

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x = −2 ± 7

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hence, solutions arex = −2 − 7 and x = −2 + 7

To complete the square we need to add b2 = +4 to both sides.

b)

€

3x2 − 5x −12 = 0 Divide by 3 and rearrange to x2 + bx = c

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x2 −−53

x =123

= 4For ax2 + 2bx + b2 = 0 2b =

€

−5

3

€

x2 −53

x +2536

= 4 +2536

€

x2 −53

x +2536

=16936

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x −56

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

=16936 Take √ of both sides.

Solve for x.

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x −56

=±13

6

€

x =5 ±13

6

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hence, solutions are x = −43

and x = 3

Factorise LHS.

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x =−43

or 3

To complete the square we need to

add b2 = to both sides.

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25

36

You need to take CARE!.€

hence, −5

3×

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

=25

36

For ax2 + bx + c = 0, if a ≠1 or 0, step 1 is to divide every term by ‘a’.

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ax 2

a+

bx

a+

c

a= 0

Exercise 2: Solve by complete the square method.

a) 2x2 - 5x - 4 = 0

b) 3x2 + 3x - 5 = 0

c) 2x2 - 7x + 1 = 0

d) 5x2 + 4x - 2 = 0

e) 3x2 - 10x + 2 = 0

€

ans : x =5 ± 57

4

€

ans : x =−3± 69

6

€

ans : x =7 ± 41

4

€

ans : x =−2 ± 14

5

€

ans : x =5 ± 19

3