When You Square Each Side Of An Equation, Is The Resulting Equation Equivalent To The Original?
When you square each side of an equation, is the resulting equation equivalent to the original equation?
If you square both sides of an equation, do you get an equivalent equation?
The answer is no. Squaring both sides of an equation will occasionally introduce an "extraneous" solution; a solution for the modified equation that is not a solution to the original equation.
These occur often when dealing with logarithms and certain trigonometric functions. They can also occur when dealing with radical equations (or equations with rational exponents) amongst other types.
(1) ` ``sqrt(x)=-3 ` : this equation clearly has no solutions in the real numbers, as the square root of a real number is nonnegative if it exists. But squaring both sides yields x=9. But x=9 is not a solution to the original equation as `sqrt(9)=3 !=-3 ` .
(2)` `` `` `` ` `sqrt(x)=sqrt(x^2-2) ` : squaring both sides yields `x=x^2-2 ==> x^2-x-2=0 ==> x=2,-1 ` . x=2 is a solution to the original equation, but x=-1 (in the real numbers) is not since ` sqrt(-1) ` is not real.
(3) x+1=x-1: squaring both sides yields `x^2+2x+1=x^2-2x+1 ==> 4x=0 ==> x=0 ` . But x=0 is not a solution to the original "equation". (The original equation has no solutions.)
Equivalent equations have the same solutions.
x+3=5 has the solution x=2.
`(x+3)^2=5^2 ` has the solutions x=2 and x=-8. Since the solutions sets are not the same, the equations cannot be equivalent.
Equivalent equations have the same solution set.
x+2=5 has the solution x=3.
`(x+2)^2=5^2 ` has the solution set x=3 and x=-7. Since the solution sets are not the same, the equations are not equivalent.
Please note that the definition of equivalent equations are equations with the same solution set.