In this chapter, we will be learning about quadratic equations including how to solve it.

Quadratic equations are equations wherein the variables are squared (x2). Thus, the name “quad”.

The “standard” equation looks like this:

ax2 + bx + c = 0

where, a, b, and c are coefficients meaning you know those numbers except that a should not be or equal to 0

However, quadratic equations “disguise” themselves into this order:

ax2 + bx = c

ax2 = bx + c

x2 + c = bx

regardless of its placement, always remember that it should have a squared variable in order for it to be called quadratic.

There are 2 ways to solve quadratic equations.

1. Factoring
2. Using the Quadratic Formula
   
          X = - b ± √ b2 – 4(a)(c)
                            2(a)
   Also remember that quadratic equations always have 2 answers.
   Let us try the examples and solve it using the factoring technique.
   

Examples:

1. 2x2 + 5x + 2 = 0
   (x + 2) (2x + 1) = 0
   X1
   X + 2 = 0
   X + 2 – 2 = 0 – 2
   
   X2
   2x + 1 = 0
   2x + 1 – 1 = 0 – 1
   2x = -1
   2x/2 = -1/2
   X = -1/2
   X = -2


2. X2 + 7x + 10 = 0
   (x + 5) (x + 2) = 0
   
   X1
   X + 5 = 0
   X + 5 – 5 = 0 – 5
   X = -5
   
   X2
   X + 2 = 0
   X + 2 – 2 = 0 – 2
   X = -2


3. 8x2 + 24x + 16 = 0
   (4x + 8) (2x + 2) = 0
   
   X­1
   4x + 8 = 0
   4x + 8 – 8 = 0 – 8
   4x = -8
   4x/4 = -8/4
   X = -2
   X2 2x + 2 = 0 2x + 2 -2 = 0 – 2 2x = -2 2x/2 = -2/2 X = -1
   
   
4. 5x2 – 17x + 6 = 0 (5x – 2) (x – 3) = 0
   X1
   5x – 2 = 0
   5x – 2 + 2 = 0 + 2
   5x = 2
   5x/5 = 2/5
   X = 2/5


5. 9x2 + 9x – 4 = 0
   (3x + 4) (3x – 1) = 0
   
   X1
   3x + 4 = 0
   3x + 4 – 4 = 0 – 4
   3x = -4
   3x/3 = -4/3
   X = -4/3
   X= -1 1/3
   
   X2
   3x – 1 = 0
   3x -1 + 1 = 0 + 1
   3x = 1
   3x/3 = 1/3
   X = 1/3
   
   

Now let us use the quadratic formula to solve the same sample equations that were given above and see if we come up with the same answers.

Examples:

1. 2x2 + 5x + 2 = 0
   
   X = - b ± √ b2 – 4(a)(c)
                   2(a)
   = - 5 ± √ 52 – 4 (2)(2)
                   2 (2)
   = - 5 ± √ 25 -16
              4
   = - 5 ± √ 9
           4
   = - 5 ± 3
           4
   
   X1 = - 5 - 3
             4
         = -8/4
         = -2
   
   X2 = - 5 + 3
               4
         = -2/4
         = -1/2
   
2. X2 + 7x + 10 = 0
   
   X = - b ± √ b2 – 4(a)(c)
                      2(a)
   = - 7 ± √ 72 – 4(1)(10)
                   2(1)
   = - 7 ± √ 49 – 40
                  2
   = - 7 ± √ 9
           2
   = - 7 ± 3
           2
   
   X1 = - 7 - 3
               2
         = -10/2
         = -5
   
   X2 = - 7 + 3
                2
         = -4/2
         = -2
   
   
3. 8x2 + 24x + 16 = 0
   
   X = - b ± √ b2 – 4(a)(c)
                   2(a)
   = - 24 ± √ 242 – 4(8)(16)
                   2(8)
   = - 24 ± √ 576 – 512
                   16
   = - 24 ± √ 64
           16
   = - 24 ± 8
           16
   
   X1 = - 24 - 8
               16
         = -32/16
         = -2
   X2 = - 24 + 8
                16
           = -16/16
           = -1
   
   
   
   
4. 5x2 – 17x + 6 = 0 X = - b ± √ b2 – 4(a)(c)
                                                     2(a)
   = - (-17) ± √ (-17)2 – 4(5)(6)
                   2(5)
   = - (-17) ± √ 289 – 120
                   10
   = - (-17) ± √ 169
           10
   = - (-17) ± 13
           10
   = 17 ± 13
          10
   
   X1 = 17 - 13
             10
         = 4/10
         = 2/5
   
   X2 = 17 + 13
               10
           = 30/10
           = 3
   
   
5. 9x2 + 9x – 4 = 0
   X = - b ± √ b2 – 4(a)(c)
                   2(a)
   = - 9 ± √ 92 – 4(9)(-4)
                   2(9)
   = - 9 ± √ 81 + 144
                   18
   = - 9 ± √ 225
           18
   = - 9 ± 15
           18
   
   X1 = - 9 - 15
               18
           = -24/18
           = -4/3
           = -1  1/3
   X2 = - 9 + 15
               18
           = 6/18
           = 1/3