how to factor terms?

Question

as i'm reading a paper a paper "An Underdetermined Linear System for GPS" By Dan Kalman

and solving an equation ,and as i'm not good in math i missed there ,in factoring of the following equation: $$t=-0.047^2\cdot2.4^2+0.047^2\cdot19.9^2$$ how can i facor the similar terms ,i mean simplify the equation

is this equation equals to,

$$t=-0.047^2(2.4^2+19.9^2)$$ or $$t=0.047^2(19.9^2-2.4^2)$$

which one is correct and why.

Answer 1

$$\begin{array}{c l} t & =-0.047^2\cdot2.4^2+0.047^2\cdot19.9^2 && (1) \\ & = 0.047^2\cdot(-2.4^2)+0.047^2\cdot19.9^2 && (2) \\ & = 0.047^2\big(-2.4^2+19.9^2 \big) && (3) \\ & = 0.047^2\big(19.9^2-2.4^2\big) && (4) \end{array}$$

Remark. Keep in mind $(-2.4^2)$ and $(-2.4)^2$ are two different things.

• $(1)\to(2)$: By commutativity, $-ab=(-1)ab=a(-1)b=a(-b)$. Above, $a=0.047^2$, $b=2.4^2$.
• $(2)\to(3)$: By distributivity, $ab+ac=a(b+c)$. Above, $a=0.047^2$, $b=-2.4^2$, $c=19.9^2$.
• $(3)\to(4)$: By commutativity, $-a+b=b+(-a)=b-a$. Above, $a=-2.4^2,b=19.9^2$.

Answer 2

First factorization is incorrect: the sign is wrong in the second summand. If you multiply it out, you get: $$-0.047^2(2.4^2+19.9^2) = (-0.047^2)\cdot 2.4^2 + (-0.047^2)\cdot 19.9^2 = -0.047^2\cdot 2.4^2 - 0.047^2\cdot 19.9^2,$$ which is not the same thing you had before. Instead, it should be $$-0.047^2(2.4^2-19.9^2)$$

The second factorization is correct. It comes from factoring out $0.047^2$ (instead of $-0.047^2$) and then reordering: $$-0.047^2\cdot 2.4^2 + 0.047^2\cdot 19.9^2 = 0.047^2(-2.4^2 + 19.9^2) = 0.047^2(19.9^2 - 2.4^2).$$