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 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 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