Is it possible to solve for $$(a-b)^2$$
by knowing only the following information: $$(r_1a-r_2b)^2 = C$$ $$r_1$$ $$r_2$$ $$C$$
Solve $(a-b)^2$ with $(r_1a-r_2b)^2$
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linear-algebra
problem-solving
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4No, for example if we have $(3a + 2b)^2 = 144$, we can have $a=2, b=3$ or $a=0, b=6$ – 2017-01-11
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2@TCiur If we want the $r_i$ to be positive, we can make $(3\cdot 8-2\cdot6)^2=(3\cdot6-2\cdot3)^2=144$, but $(8-6)^2\neq (6-3)^2$. – 2017-01-11
1 Answers
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Let $x=a-b$. So we are trying to find $x^2$.
We know that $(r_1a-r_2b)^2=C$. Next substitute in for $x$:
$$(r_1(x+b)-r_2b)^2=C$$
$$r_1(x+b)-r_2b=\pm\sqrt{C}$$
$$x+b-\frac{r_2b}{r_1}=\pm\frac{\sqrt{C}}{r_1}$$
$$x=-b+\frac{r_2b}{r_1}\pm\frac{\sqrt{C}}{r_1}$$
$$x^2=\left(-b+\frac{r_2b}{r_1}\pm\frac{\sqrt{C}}{r_1}\right)^2$$
Note that $x^2$ depends upon the value of $b$ which we do not know so it can not be found from the information given.