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3x^2 - 2x - 3 = 0 ----(1)
\n" ); document.write( "Multiplying by 3
\n" ); document.write( "9x^2-6x-9 =0 ----(2) (zero mulitplied by anything is zero)
\n" ); document.write( "(3x)^2- 2X(3x)-9= 0
\n" ); document.write( "That is (3x)^2- 2X(3x)X(1)= 9
\n" ); document.write( "[The left hand side is like (a)^2 -2ab with a =(3x) and b = 1. we require b^2 to make it a perfect square: (a)^2 -2ab +b^2 = (a-b)^2 ]
\n" ); document.write( "Adding b^2 = 1^2 = 1 on both the sides
\n" ); document.write( "(3x)^2- 2X(3x)X (1) +1 = 9 +1
\n" ); document.write( "(3x-1)^2 = 10
\n" ); document.write( "Taking the sqrt on both the sides
\n" ); document.write( "(3x-1) = +or- sqrt(10)
\n" ); document.write( "3x = 1+or- sqrt(10)
\n" ); document.write( "x = (1/3)times[1+or- sqrt(10)]
\n" ); document.write( "Answer: x = (1/3)(1+sqrt10) and x= (1/3)(1-sqrt10)
\n" ); document.write( "Verification: x = (1/3)(1+sqrt10) in (1) implies
\n" ); document.write( "LHS = 3x^2 - 2x - 3
\n" ); document.write( "= 3X(1/9)(1+10+2sqrt10)-2X(1/3)X(1+sqrt10)-3
\n" ); document.write( "=(1/3)[11+2sqrt10-2(1+sqrt10)]-3
\n" ); document.write( "=(1/3)[11+2sqrt10-2-2sqrt10)]- 3
\n" ); document.write( "=(1/3)(11-2) - 3
\n" ); document.write( "=(1/3)X(9) - 3
\n" ); document.write( "=3-3 = 0 =RHS
\n" ); document.write( "Therefore x = (1/3)(1+sqrt10) is correct
\n" ); document.write( "Since surd roots occur in conjuate pairs in equations
\n" ); document.write( "there is no need for us to test the validity of the other root.
\n" ); document.write( "Note: Why did we multiply through out by 3 in the beginning?
\n" ); document.write( "Abs: To make the first term a perfect square
\n" ); document.write( "to play the role of a in the perfect square (a-b)^2\r
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