Chapter 2 Worked Examples: Quadratic functions and equations
- Created by: Milly
- Created on: 10-04-13 12:29
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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