Question

answer this I wanna a see who is smart hear and want correct answer and want answer with explanision​

Answer 1

Step-by-step explanation:

Given :-

x+y+z = 14

x²+y²+z² = 91

y² = zx

To find:-

Find the possible values of x,y,and z ?

Solution:-

Given that :

x+y+z = 14 ------------(1)

x²+y²+z² = 91 --------(2)

y² = zx -----------------(3)

On squaring both sides of the equation (1) then

=> (x+y+z)² = (14)²

=> x²+y²+z²+2xy+2yz+2zx = 196

=> x²+y²+z²+2(xy+yz+zx) = 196

=> 91 +2(xy+yz+zx) = 196

=> 2(xy + yz + zx ) = 196-91

=> 2(xy + yz + zx ) = 105

=>2(xy+yz+y²) = 105 (from (3))

=> 2[y(x+z)+y²] = 105

=>2[y(14-y)+y²] = 105

Since x+y+z = 14 => x+z = 14-y

=> 2(14y-y²+y²) = 105

=>2(14y) = 105

=> 28y = 105

=> y = 105/28

=> y = 15/4

On Substituting the value of y in (1)

=> x+z+(15/4) = 14

=> x+z = 14-(15/4)

=> x+z = (56-15)/4

=> x+z = 41/4 --------(4)

and

On Substituting the value of y in (3) then

=> zx = (15/4)²

=> zx = 225/16 ------(5)

=> x+(225/16x) = 41/4

=> (16x²+225)/16x = 41/4

On applying cross multiplication then

=> (16x²+225)×4 = 16x×41

=> 16x²+225 = 4x×41

=> 16x²+225 = 164x

=> 16x²+225-164x = 0

=> 16x²-164x+225 = 0

Using Quadratic formula

x = [-b±√(b²-4ac)]/2a

We have , a = 16 , b = -164 , c = 225

=> x = [164±√{(-164)²-4(16)(225)}]/2(16)

=> x = [164+√(26896-21150)/32

=> x = (164+√5746)/32

=> x = (164+75.8)/32

Since √5746 = 75.80023...

=> x = 239.8/32

=>x = 2398/320

=> x = 1199/160

On Substituting the value of x in (4)

(1199/160)+z = 41/4

=> z = (41/4)-(1199/160)

=>z = (1640-1199)/160

=> z = 441/160

So,

x = 1199/160 , y = 15/4 , z = 441/160

Answer:-

The possible values of x,y and z are 1199/160, 15/4 , 441/160 respectively.

Check:-

x+y+z =

(1199/160)+(15/4)+(441/160)

=> (1199+600+441)/160

=>2240/160

=> 14

x+y+z = 14

Verified the given relations in the given problem.

Used formulae:-

• (x+y+z)² = x²+y²+z²+2xy+2yz+2zx
• Quadratic formula for finding the roots for the equation is ax²+bx+c = 0 is x = [-b±√(b²-4ac)]/2a