As pka said, multiplying by 3 gives
x 2 − 28 x + 27 = ( x − 1 ) ( x − 27 ) \displaystyle x^2- 28x+ 27= (x- 1)(x- 27) x 2 − 2 8 x + 2 7 = ( x − 1 ) ( x − 2 7 )
But the quadratic formula certainly ought to give them too!
The "discriminant" is
b 2 − 4 a c = ( 28 3 ) 2 − 4 ( 1 3 ) ( 9 ) = 784 9 − 36 3 = 784 9 − 108 9 \displaystyle b^2- 4ac= \left(\frac{28}{3}\right)^2- 4\left(\frac{1}{3}\right)(9)= \frac{784}{9}- \frac{36}{3}= \frac{784}{9}- \frac{108}{9} b 2 − 4 a c = ( 3 2 8 ) 2 − 4 ( 3 1 ) ( 9 ) = 9 7 8 4 − 3 3 6 = 9 7 8 4 − 9 1 0 8
= 676 9 = ( 26 3 ) 2 \displaystyle = \frac{676}{9}= \left(\frac{26}{3}\right)^2 = 9 6 7 6 = ( 3 2 6 ) 2
So
− b ± b 2 − 4 a c 2 a = − 28 3 ± 26 3 2 ( 1 / 3 ) \displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}= \frac{-\frac{28}{3}\pm\frac{26}{3}}{2(1/3)} 2 a − b ± b 2 − 4 a c = 2 ( 1 / 3 ) − 3 2 8 ± 3 2 6
Taking the "+",
x = ( − 28 3 + 26 3 ) ( 3 / 2 ) = ( 2 3 ) ( 3 2 ) = 1 \displaystyle x= (-\frac{28}{3}+ \frac{26}{3})(3/2)= \left(\frac{2}{3}\right)\left(\frac{3}{2}\right)= 1 x = ( − 3 2 8 + 3 2 6 ) ( 3 / 2 ) = ( 3 2 ) ( 2 3 ) = 1 .
Taking the '-",
x = ( − 28 3 − 26 3 ) ( 3 / 2 ) = ( 54 3 ) ( 3 2 ) = 27 \displaystyle x= (-\frac{28}{3}- \frac{26}{3})(3/2)= \left(\frac{54}{3}\right)\left(\frac{3}{2}\right)= 27 x = ( − 3 2 8 − 3 2 6 ) ( 3 / 2 ) = ( 3 5 4 ) ( 2 3 ) = 2 7 .
Last edited: Apr 15, 2013