Quadratic Equation – Part 9

Type 9: x^{2}-5x-336

Finding roots for the above problem is difficult. So here is the easy method to solve such kind of problems.

Step 1: Find the LCM for the constant number 336

Observe the below diagram

Step 2: Use the formula – Multiply all even factors and all odd factors separately.

Here the even factor is 2 and the odd factor is 7 and 3

(2*2*2*2), (7*3)

16, -21 (When we add these two we will get “-5”)

Step 3: When we add these two factors we will get -5 and when we multiply, we get 336

16-21=-5           16*21=336

The roots are 16, -21.

Example 2:  x^{2}-239x-972

Step 1: Find the LCM for the constant number 336

Observe the below diagram

Step 2: Use the formula – Multiply all even factors and all odd factors separately

Here the even factor is 4 and the odd factor is 3

(4) ; (3*3*3*3*3)

4, -243

Step 3: When we add  these two factors we will get -239 and when we multiply we will get 972

4-243=-239; 4*243=239

The roots are 4, -243

Quadratic Equations – Part 7

Type 7:  2x^{2}-(4+\sqrt{13})x+2\sqrt{13}

Basic method : 2x^{2}-4x-\sqrt{13}x+2\sqrt{13}

2x(x-2)-\sqrt{13}(x-2)

(2x-\sqrt{13})(x-2)

2x-\sqrt{13}=0            x-2=0

x=\frac{\sqrt{13}}{2}                          x=2

Shortcut method:

Step 1: Consider the middle term  (4+\sqrt{13})

Step 2: Split the term into 4 and \sqrt{13}

Step 3: Divide  4 and \sqrt{13}  by the first term 2.

\frac{4}{2},\frac{\sqrt{13}}{2}=2,\frac{\sqrt{13}}{2}

The roots of the given equation are  2,\frac{\sqrt{13}}{2}

10y^{2}-(18+5\sqrt{13})y+9\sqrt{13}

Step 1: Consider the middle term  (18+5\sqrt{13})

Step 2: Split the term into 18 and 5\sqrt{13}

Step 3: Divide  18 and 5\sqrt{13}  by the first term 10

\frac{18}{10},\frac{5\sqrt{13}}{10}=1.8,\frac{\sqrt{13}}{2}

The roots of the given equation are  1.8,\frac{\sqrt{13}}{2}

The final answer is x\geq y

Quadratic Equation – Part 6

Problem Statement: x^{2}-7\sqrt{3}x+36 ; y^{2}-12\sqrt{2}y+70 which is greater x or y?

Solution: When you find any root digit in the equation it is difficult to solve such a problem. So here is the shortcut method is

x^{2}-7\sqrt{3}x+36

Step 1: Just divide the constant  36 with the middle number which in the root

\frac{36}{3}=12

Step 2: Forgot,^{\sqrt{3}} now the equation is

x^{2}-7x+12

Step 3: Follow the procedure which is shown in part 1

Observe the diagram as shown below

Step 4: Finally add ^{\sqrt{3}} to the resultant roots

3\sqrt{3},4\sqrt{3}

y^{2}-12\sqrt{2}y+70

Step 1: Just divide the constant  70 with the middle number which is the root

\frac{70}{2}=35

Step 2: Forgot, ^{\sqrt{2}} now the equation is

y^{2}-12y+35

Step 3: Follow the procedure which is shown in part 1

Observe the diagram as shown below

Step 4: Finally add ^{\sqrt{2}} to the resultant roots

7\sqrt{2},5\sqrt{2}

Finally, the answer is y>x.

Quadratic Equations – Part 5

Type: Inequalities

When we got such cases like below in the given problem the answer is cannot be determined.

  • (-,-) and (+,-) 
  • (-,-) and (-,-)
  • (+,-) and (+,-)

Example 1: x^{2}+x-20 ; y^{2}-y-30 which is a greater value x or y from both equations?

It is matched with the first case so the answer is cannot be determined or relationship does not exist.

Example 2x^{2}+x-206y^{2}+y-1 which is a greater value x or y from both equations?

It is matched with the third case so the answer is cannot be determined or relationship does not exist.

Example 312x^{2}-2x-4 y^{2}-y-30

It is matched with the second case so the answer is cannot be determined or relationship does not exist.

Example 4: If you find  (-, +)  and (+,+) pair in the given problem the answer is (-,+)  >  (+,+)

Example: y^{2}+8y+15 x^{2}-8x+15

The answer is X>Y

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

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 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 chance 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