Equation Problem

[jpanknin]

Hi everyone, I have hit a conundrum with a problem (see screenshot below). I can't figure out why I'm getting 2 = 1 for Method 3. I'm trying a lot of different ways to manipulate equations/expressions and this one has stumped me. I know I'm not solving for x in Method 3, but I thought both sides should be equal since they both equal 0.

 

[jonah2.0]

Beer drenched reaction follows.

Hi everyone, I have hit a conundrum with a problem (see screenshot below). I can't figure out why I'm getting 2 = 1 for Method 3. I'm trying a lot of different ways to manipulate equations/expressions and this one has stumped me. I know I'm not solving for x in Method 3, but I thought both sides should be equal since they both equal 0.

View attachment 39853

Let us ask the the Google AI god:

 

[Dr.Peterson]

Hi everyone, I have hit a conundrum with a problem (see screenshot below). I can't figure out why I'm getting 2 = 1 for Method 3. I'm trying a lot of different ways to manipulate equations/expressions and this one has stumped me. I know I'm not solving for x in Method 3, but I thought both sides should be equal since they both equal 0.

View attachment 39853

First, it's worth observing that the first equation in your system is just second the first, so it is redundant. So you might as well just solve the second alone. That's what you end up doing in the first two methods anyway.

As for the third method, surely you have been taught not to divide by an expression that might be zero. As the AI said, when you are tempted to do that, you should use factoring instead; what it did is essentially your first method, except that you used a square root instead of factoring.

So what actually happened in method 3? You discovered the fact I mentioned, that the first expression is twice the second. The only way one expression can equal its own double is if it is zero; so when you assume it is not zero (in order to divide by it), you find that the equation is impossible. And that's what 2=1 tells you. When you divide by such an expression, you have to supplement your work by checking when it is zero; any solutions to that will be solutions to the original equation.

 

[jpanknin]

Ah, I see. The only place that [imath]2x^2-50[/imath] and [imath]x^2-25[/imath] are equal is at x = 5 or x = -5, which would then be division by zero. So by setting each expression to 0 I guaranteed division by zero. Makes sense. Thank you very much.

 

[Dr.Peterson]

Ah, I see. The only place that [imath]2x^2-50[/imath] and [imath]x^2-25[/imath] are equal is at x = 5 or x = -5, which would then be division by zero. So by setting each expression to 0 I guaranteed division by zero. Makes sense. Thank you very much.

I wouldn't quite put it that way. Since your original problem, as stated (equations 1 and 2), is to find when both expressions are zero, and the two equations are equivalent, the solution is the same as the solution to either equation, namely x=5 or -5.

By setting the two expressions equal as part of solving the system, you lost that information; and in trying to solve by dividing, you accidentally divided by zero.