Chapter 2 Worked Examples: Quadratic functions and equations

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Example 1

Solve these equations by factorisation 

a) x^2 - 5x -14 =0

b) 3x^2 +2x -5 =0

c) 4x^2 +4x +1 =0

Answers

a) (x-7)(x+2)=0

x-7=0 x=7

x+2=0 x=-2

x = 7 or -2

b) (3x+5)(x-1)=0

3x+5=0 x=-5/3

x-1=0 x=1

x=-5/3 or 1

c) (2x+1)(2x+1)=0

(2x+1)^2=0

2x+1=0

x=-1/2

Example 2

Solve these equations by factorisation

a) 8x^2 -5 =10x-2

b) x(x+10)=4(x-2)

Answers

a) 8x^2-10x-3=0

(4x+1)(2x-3)=0

4x+1=0 x=-1/4

2x-3=0 x=1.5

x= -1/4 or 1.5

b) x^2+10x=4x-8

x^2+6x+8=0

(x+4)(x+2)=0

x+4=0 x=-4

x+2=0 x=-2

x= -4 or -2

Example 3

Solve these equations without using the GDC

a) x^2+14x+49=5

b) x^2 -6x +9=6

Answers

a)(x+7)^2=5

x+7=± 5

x=-7± 5

b) (x-3)^2=6

x-3=± 6

x=3± 6

Example 4

Solve each equation by completing the square 

a) x^2+10x=6

b) x^2-12x=3

c) x^2-3x-1=0

Answers

a) x^2+10x+25=6+25

(x+5)^2=31

x+5=± 31

x=-5± 31

b) x^2-12x+36=3+36

(x-6)^2=39

x-6=±√39

x6±√39

c) x^2-3x=1

x^2-3x+2.25=1+2.25

(x-1.5)^2=3.25

x-1.5=±√3.25

x=1.5±√3.25

  • Divide the coefficient by 2 and square it, then add this to both sides of the equation 

Example 5

Solve each of these equations by completing the square 

a) 2x^2 +8x=6

b) 3x^2 -15x=2

Answers

a) x^2+4x=3

x^2+4x+4=3+4

(x+2)^2=7

x+2=±√7

x=-2±√7

b) x^2-5x=2/3

x^2-5x +6.25=2/3 +6.25

(x-2.25)^2=6.917

x-2.25=±√6.917

x=2.25±√6.917

Example 6

Solve each equation using the quadratic formula 

a) x^2 + 4x - 6 = 0

b) 2x^2 - 3x = 7

c) 3x^2 = 7x + 6

Answers

a) x = (-4±√4^2 -(4*1*-6))/2(1)

x = (-4±√40)/2

x = (-4±2√10)/2

x = -2±√10

b) 2x^2 - 3x - 7 = 0 

  x = (3±√-3^2 -(4*2*-7))/2(2)

x = (3±√65)/4

c) 3x^2  - 7x - 6 = 0 

 x = (7±√-7^2 -(4*3*-6))/2(3)

x = (7±√121)/6

x = (7±11)/6

x = -2/3 or 3 

Example 7 

The sum of the squares of two consecutive integers is 613. Find the two integers. 

Answer

x^2 + (x+1)^2 = 613

x^2 + x^2 +2x + 1 =

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