Quadratic Equations – Part 3

Type 3:  42x^{2}+97x+56

Basic method:

To solve this problem in basic method we can use one formula

i.e.  \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}    here a=42    b=97     c=56

By using this formula we can not solve this problem in less time. So to solve this problem in less time there an alternative method as shown below.

Shortcut Method:

Step 1: Find the factors for first term 42

The factors are 2,3,6,7,42,1

6,7 are the appropriate factors for this case (See the below diagram for more details)

Step 2: Find the factors for constant 56

The factors are 2,3,7,8,56,1

7,8 factors are the appropriate factors for this case (See the below diagram for more details)

Step 3: Select 7 in the first one and 7 in the second one and multiply both

7*7=49

Step 4: Select 6 in the first one and 8 in the second one and multiply both

6*8=48

Step 5: Add both 49+48=97 which is equal to middle number i.e.97

Step 6: Divide the,\frac{49}{42} \frac{48}{42}

The result after the division –   \frac{7}{6} , \frac{8}{7}

Step 7: Change the sign -\frac{7}{6} , -\frac{8}{7}

 

Observe the diagram for the given equation

Quadratic Equations- Part 2

Type 2:   2X^{2}+8X+8

Basic method: 2X^{2}+8X+8

2x^{2}+4x+4x+8

2x(x+2)+4(x+2)

(2x+4)(x+2)

2x=-4            x=-2

x=-\frac{4}{2}

x=-2

Shortcut method:

  • Step 1: Multiply the first number “2” and the constant 8
    • 2*8=16
  • Step 2: Find the factors for 16
    • 2,4,8,16,1 are the factors.
  • Step 3: Select the factors such that when we add we should get the middle number (8 in the given equation) and when we multiply we should get constant (16).
    • 4 and 4 are such factors which will give sum as 8 and multiplication as 16.
    • If we take 1 and 6, multiplication is 6  but the sum is 7 which is not equal to 5.
    • So the appropriate factors for the given equation are 3 and 2.
  • Step 4: Change the sign for the  factors
    • Since it is ”+8x” in the given equation we need to change the sign for the factors  i.e. -4,-4
    • If it is “-8x” in the equation we need to change the sign for the factor as 4, 4.
  • Step 4: Divide the resultant factors by the first number 2
  • -\frac{4}{2},-\frac{4}{2}
  • -2,-2  are the final roots.

Observe the diagram below

Quadratic Equations – Part 1

Type 1: x^{2}+5x+6= 0,  Find the roots for the equation?

Basic method:

x^{2}+5x+6

x^{2}+3x+2x+6

x(x+3)+2(x+3)

(x+2)(x+3)

x+2=0            x+3=0

x=-2                x=-3

Short Cut Method:

  • Step1: Find the factors for constant (In the given equation it is 6)
    • 2,3,6,1 are the factors for 6
  • Step 2: Select the factors such that when we  add we should get middle number (5 in the given equation) and when we multiply we should get constant (6).
    • 2 and 3 are such factors which will give sum as 5 and multiplication as 6.
    • If we take 1 and 6, multiplication is 6  but the sum is 7 which is not equal to 5.
    • So the appropriate factors for the given equation are 3 and 2.
  • Step 3: Change the sign  for the  factors
    • Since it is ”+5x” in the given equation we need to change the sign for the factors  i.e. -3,-2
    • If it is “-5x” in the equation we need to change the sign for the factor as 3, 2.

Note: 2 and 3 are co-primes. So we should select co-prime factors.

Observe the below diagram