A rectangular garden is 12ft by 8ft. The same length is added around the whole garden. The new area is 212ft. How much was added to each side?

Question

 know that x has been added to each side and it is the same amour. We need to find x.

Answer 1

original rect. garden 

12ft x 8ft 

 

add X to the length, and add X to the width

 

new dimensions of length and width

12+X,  8+X 

 

AREA is L x W and = 212, so:

(12+X)(8+X) = 212  

 

EXPAND LEFT SIDE:

96 + 8X + 12X + X² =  212

 

ADD X TERMS AND REARRANGE:

X² + 20X + 96 = 212    OR:

 

1)   X² + 20X - 116 = 0         THIS IS THE QUADRATIC TO BE SOLVED. 

 

SOLVE FOR X, FACTORING DOESN'T APPEAR TO WORK, SO WE WILL COMPLETE THE SQUARE

FIRST ADD 116 TO EACH SIDE:

  

X² + 20X = 116

 

TO COMPLETE THE SQUARE, ADD (B/2)² TO EACH SIDE OF EQN 1:  (NOTE:  A=1,  B=20)

 

X² + 20X + 10² = 116 + 10²  = 216

 

(X+10)²  = 216

 

TAKE SQUARE ROOT OF EACH SIDE  (NOTE: WHENEVER YOU TAKE A SQRT, NEED THE +/- )

 

X+10  = +/- SQRT(216) =  +/- 6*SQRT(6)        NOW SUBT 10 FROM EACH SIDE

 

X = -10 +/- 6*SQRT(6)        NOW, THE ONLY SOLN IS THE (+)  (X CAN'T BE NEGATIVE)

 

X = -10 + 6*SQRT(6)   = 4.6969    or about 4.7

 

(IF YOU PLOT QUADRATIC EQN. 1, YOU'LL SEE THE SOLUTION, WHERE IT CROSSES THE X AXIS AT THIS VALUE)

 

THE NEW DIMENSIONS ARE:  (12+4.7) AND (8+4.7)  = 16.7 and 12.7

 

(note:  16.7 x 12.7 = 212.09, the 0.09 extra from rounding to nearest decimal 4.7 for X)