The roots are 3 and -k
so the base length is |3 - - k| = |3+K|
The height is given by the y intercept which is |0.5 * -3 * k| = | -1.5k |
A=0.5∗|3+k|∗|−1.5k|A=0.5∗1.5∗|k|∗|3+k|but we want this to equal 1.51.5=0.5∗1.5∗|k|∗|3+k|1=0.5∗|k|∗|3+k|2=|k|∗|3+k| Ifk<−3then2=−k∗−(3+k)2=+k(3+k)...k=−3±√172k=k=−3−√172k≈−3.56
\(If\;\;-3
Ifk>0then2=k∗(3+k)...k=−3±√172k=−3+√172k≈0.56
So k=−2,−1,−3−√172,−3+√172would all work
It would be a good idea to test each of these for validity.
LaTex:
A=0.5*|3+k|*|-1.5k|\\
A=0.5*1.5*|k|*|3+k|\\
\text{but we want this to equal } 1.5\\
1.5=0.5*1.5*|k|*|3+k|\\
1=0.5*|k|*|3+k|\\
2=|k|*|3+k|\\~\\
If\;\;k<-3\;\;then\\
2=-k*-(3+k)\\
2=+k(3+k)\\
...\\
k=\frac{-3\pm\sqrt{17}}{2}\\
k=k=\frac{-3-\sqrt{17}}{2}\\
k\approx -3.56
If\;\;-3 2=-k*(3+k)\\
-2=k(3+k)\\
...\\
k=\frac{-3\pm\sqrt{17}}{2}\\
k=\frac{-3+\sqrt{17}}{2}\\
k\approx 0.56
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