Quadratic Equations by Example 

-- Part 1.3 --

  ax2 + bx + c = 0

Solve for x:   6x2 + 3x = 0        Factoring Method

Solve

Step

Check 

   6x2 + 3x = 0
3x(2x + 1) = 0
3x = 0 or 2x + 1 = 0
3x = 0 gives x = 0/3 = 0
or 
2x + 1 = 0 gives 2x = -1 
x = -1/2
{-1/2, 0}

Factor, if possible.

Zero Property of Multiplication
If ab = 0 then a = 0 or b = 0

Solve each linear equation separately

Solution Set Notation

If x = 0, then
6(0)2 + 3(0) = 0 
    0 + 0 = 0 

If x = -1/2, then
6(-1/2)2 + 3(-1/2) = 0  
    6(1/4) – 3/2 = 0 
3/2 – 3/2 = 0

Solve for x:   8x2 + 2x = 0

[Solution]

    Solve for x:   2x2 + x = 6     Factoring Method

Solve

Step

Check 

  2x2 + x – 6 =  0
(2x – 3)(x + 2) = 0
2x – 3 = 0 or x + 2 = 0

2x – 3 = 0 gives 2x = 3
x = 3/2
x + 2 = 0 gives x = -2

{-2, 3/2}

Factor, if possible. You must equation equal to 0 first.

Zero Property of Multiplication
If ab = 0 then a = 0 or b = 0

Solve each linear equation separately

Solution Set Notation

If x = -2, then
2(-2)2 + (-2) = 6 
    8 – 2 = 6 

If x = 3/2, then
2(3/2)2 + (3/2) = 6  
    2(9/4) + 3/2 = 0 
9/2 + 3/2 = 6
12/2 = 6

Solve for x:  3x2 + x = 2 

[Solution]

    Solve for x:  (x + 2)2 = 3          
   Square Root Method

Solve

Step

Check 

 (x + 2)2 = 3   

 Original equation

If x  = -0.268, then

  

 Take square root of both sides

 (-0.268 + 2)2 = 3

Subtract 2 from both sides

    (1.732)2 = 3
2.999824 = 3
 **Rounding error

 (exact)
or
 (exact)

 There are 2 distinct answers.

Approximations...
x -2 + 1.732 = -0.268
x -2 – 1.732 = -3.732

If x  = -3.732, then
 (-3.732 + 2)2 = 3
    (-1.732)2 = 3
2.999824 = 3
**Rounding error
**You may notice that the approximations did equal 3 exactly.  Recall that the square root of 3 is an irrational number and can not be expressed exactly as a decimal. You can get closer by using more decimal places.

  Solve for x:  (x – 4)2 = 5    

[Solution]

      Solve for x:  (5x – 2)2 = 4

Solve

Step

Check 

(5x – 2)2 = 4

 Original equation

If x  = 4/5, then

  5x – 2 = ± 2

 Take square root of both sides

 (5(4/5) – 2)2 = 4

5x – 2 = 2  or 5x – 2 = -2

 Solve each equation separately

 (4 – 2)2 = 4

5x – 2 = 2
5x = 4
x = 4/5

 Solve first equation

(2)2 = 4

If x  = 0, then

  5x – 2 = -2
5x = 0
x = 0/5 = 0

 Solve second equation

 (5(0) – 2)2 = 4

 (- 2)2 = 4 

  Solve for x:  (3x – 1)2 = 9 

[Solution]

      Solve for x:  x2 + 10x + 3 = 0 Link - Completing the Square 

Solve

Step

Check 

x2 + 10x + 3 = 0
x2 + 10x      = -3

 Original equation
Subtract 3 from both sides

If x  = -0.3096, then
  (-0.3096)2 + 10(-0.3096) + 3 = 0

x2 + 10x+ (10/2)2  = -3 + (10/2)2

 Take half of 10, square it, add it to both sides

  .095852 – 3.096 + 3 = 0
-.00015 = 0
**Rounding error

(x + 5)2  = -3 + 25
(x + 5)2  = 22

Take square root of both sides If x  = -9.6904, then

(-9.6904)2 + 10(-9.6904) + 3 = 0

Solve each equation separately

93.90385 –  96.904 + 3 = 0
-.00015 = 0
**Rounding error

Approximations...
x -5 + 4.6904  = -0.3096
x -5 – 4.6904 = -9.6904
**You may notice that the approximations did equal 3 exactly.  Recall that the square root of 3 is an irrational number and can not be expressed exactly as a decimal. You can get closer by using more decimal places.

 Solve for x:  x2 + 6x – 1 = 0  Link - Completing the Square 

[Solution]

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Last Update 03.15.2006