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If $(a + c)(a − c) = 0$, which of the following must be true?

  1. $a = 0$,
  2. $c = 0$,
  3. $a = −c$,
  4. $a = c$,
  5. $a2 = c2$

The answer states: Try plugging in $a = 2$ and $c = 2$. This eliminates (A), (B), and (C). Now try $a = 2$ and $c = −2$. This eliminates (D). Only (E) remains.

I don't understand how the plug in numbers were chosen. Could anyone help explain this?

dustin
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Victor
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  • Looking at the multiple choice answers the problem is obviously looking for values of $a$ (totally descarding choice E). Therefore, what values of $a$ would make this product equal to zero. Hint, there is more than one. – K7PEH Nov 13 '14 at 02:39
  • You want to eliminate some choices. You arbitrarily choose $a=c\neq 0$. If you plug the value in, and find that the equation is true for those values, then you have successfully eliminated options $1,2,$ and $3$. – Sujaan Kunalan Nov 13 '14 at 02:57
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    If you meant $a^2 = c^2$ in part 5, you can indicate that by writing a^2 = c^2 between the dollar signs. – N. F. Taussig Nov 13 '14 at 02:59

2 Answers2

1

By choosing $a = 2$, $c = 2$, the author demonstrates that the equation can be satisfied when $a \neq 0$ since

$$(a + c)(a - c) = (2 + 2)(2 - 2) = 4 \cdot 0 = 0$$

Hence, $a = 2$, $c = 2$, is a counterexample to the claim that $a = 0$ must be true. There are others. For instance, $a = 1$, $c = 1$ or $a = 1$, $c= -1$. However, only one counterexample is required to demonstrate that a statement is false.

What the author is stating is that $a = 2$, $c = 2$ is also a counterexample to the claim that $b = 0$ must be true or that $a = -c$ must be true for the equation $(a + c)(a - c) = 0$ to be satisfied.

Similarly, if $a = 2$ and $c = -2$, then

$$(a + c)(a - c) = [2 + (-2)][2 - (-2)] = 0 \cdot 4 = 0$$

so $a = 2$, $c = -2$ is a counterexample to the claim that $a = c$ must be true for the equation $(a + c)(a - c) = 0$ to be satisfied. Again, there are other counterexamples such as $a = 1$, $c = -1$.

As for the remaining choice, since $(a + c)(a - c) = a^2 - c^2$, $$(a + c)(a - c) = 0 \Rightarrow a^2 - c^2 = 0 \Rightarrow a^2 = c^2$$

N. F. Taussig
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0

Another way to come to the same conclusion is to multiply the expression out: $$ (a+c)(a-c) = a^2 - c^2 $$ so if $(a+c)(a-c) = 0$ then you also have $a^2 - c^2 = 0$. Adding $c^2$ to each side shows (E) is a correct answer.

This allows you to sidestep the written answer's ad hoc choices of values which they use to eliminate the first 4 options.

Titus
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