Decimal numbers

When two decimal numbers are multiplied, the answer is 0.072. When the same two decimal numbers are added, the answer is 0.72. What are the two decimal numbers?

Correct answer:

a =  0.6
b =  0.12

Step-by-step explanation:

$$\begin{gathered} a\cdot b=0.072 \\ a+b=0.72 \\ a\cdot (0.72-a)=0.072 \\ a\cdot (0.72-a)=0.072 \\ -a^2+0.72a-0.072=0 \\ {a}^2-0.72a+0.072=0 \\ p=1;q=-0.72;r=0.072 \\ D=q^2-4pr=0.72^2-4\cdot 1\cdot 0.072=0.2304 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{0.72\pm \sqrt{0.23}}{2} \\ {a}_{1,2}=0.36\pm 0.24 \\ {a}_1=0.6 \\ {a}_2=0.12 \\ a=a_1=0.6 \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} b=0.72-a=0.72-0.6=\frac{3}{25}=0.12 \\ b=a_2 \\ \text{ Verifying Solution: } \\ m=a\cdot b=0.6\cdot 0.12=\frac{9}{125}=0.072 \\ s=a+b=0.6+0.12=\frac{18}{25}=0.72 \end{gathered}$$

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