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The area of the triangle formed by x- and y-intercepts of the parabol y=0.5(x-3)(x+k) is equal to 1.5 square units. Find all possible values of k.

I only got -2 and -1, but it showed it was wrong. Can someone help me figure this out?

 Feb 2, 2021

Best Answer 

 #2
avatar+112523 
+3

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 |

 

\(A=0.5*|3+k|*|-1.5k|\\ A=0.5*1.5*|k|*|3+k|\\ \text{but we want this to equal } 1.5\\ 1.5=0.5*1.5*|k|*|3+k|\\ 1=0.5*|k|*|3+k|\\ 2=|k|*|3+k|\\~\\ If\;\;k<-3\;\;then\\ 2=-k*-(3+k)\\ 2=+k(3+k)\\ ...\\ k=\frac{-3\pm\sqrt{17}}{2}\\ k=k=\frac{-3-\sqrt{17}}{2}\\ k\approx -3.56\)

 

 

\(If\;\;-3

 

 

\(If\;\;k>0\;\;then\\ 2=k*(3+k)\\ ...\\ k=\frac{-3\pm\sqrt{17}}{2}\\ k=\frac{-3+\sqrt{17}}{2}\\ k\approx 0.56\)

 

 

So       \(\boxed{k=-2,\;-1, \; \frac{-3-\sqrt{17}}{2},\; \frac{-3+\sqrt{17}}{2}\\ \text{would all work}}\)

 

 

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

 Feb 2, 2021
 #1
avatar+112523 
+3

I did it graphically and got one answer of   k=0.56    (probably correct to 2 decimal places)

 

 Feb 2, 2021
 #2
avatar+112523 
+3
Best Answer

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 |

 

\(A=0.5*|3+k|*|-1.5k|\\ A=0.5*1.5*|k|*|3+k|\\ \text{but we want this to equal } 1.5\\ 1.5=0.5*1.5*|k|*|3+k|\\ 1=0.5*|k|*|3+k|\\ 2=|k|*|3+k|\\~\\ If\;\;k<-3\;\;then\\ 2=-k*-(3+k)\\ 2=+k(3+k)\\ ...\\ k=\frac{-3\pm\sqrt{17}}{2}\\ k=k=\frac{-3-\sqrt{17}}{2}\\ k\approx -3.56\)

 

 

\(If\;\;-3

 

 

\(If\;\;k>0\;\;then\\ 2=k*(3+k)\\ ...\\ k=\frac{-3\pm\sqrt{17}}{2}\\ k=\frac{-3+\sqrt{17}}{2}\\ k\approx 0.56\)

 

 

So       \(\boxed{k=-2,\;-1, \; \frac{-3-\sqrt{17}}{2},\; \frac{-3+\sqrt{17}}{2}\\ \text{would all work}}\)

 

 

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

Melody Feb 2, 2021
 #3
avatar+116125 
+2

Nice work, Melody   !!!!     Impressive   !!!!

 

cool cool cool

CPhill  Feb 2, 2021
 #4
avatar+112523 
+2

Thanks Chris,

I got in a bit of a tangle while I was doing it.  But the knots came out ok :)

Melody  Feb 2, 2021
 #5
avatar+45 
+1

Thanks Melody! 

 Feb 2, 2021
 #6
avatar+112523 
+1

Some of my LaTex, (therefore some of my answer) has disappeared     sad

 Feb 3, 2021

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