SOLUTION: Solve by using the quadratic formula. The solution set is { , } or is solution set 0? {{{ 3p^2=-13p-12 }}}

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Question 290119: Solve by using the quadratic formula.
The solution set is { , } or is solution set 0?
+3p%5E2=-13p-12+
Answer by Deina(147) About Me  (Show Source):
You can put this solution on YOUR website!
First let's convert +3p%5E2+=+-13p+-12+ to a quadratic,
in the form of +ax%5E2+%2B+bx+%2B+c+=+0+ so:
+3p%5E2+=+highlight%28-13p+-12%29+ becomes:
+3p%5E2+%2B+highlight%28+13p+%2B+12%29+=+0+

Now plug those numbers into the quadratic formula,
replacing a b & c as above:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

a = 3
b = 13
c = 12
Replacing "x" with "p" highlight%28x%29+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ Becomes highlight%28p%29+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

Replacing "a" with "3" p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Ahighlight%28a%29%2Ac+%29%29%2F%282%2Ahighlight%28a%29%29+ becomes p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Ahighlight%283%29%2Ac+%29%29%2F%282%2Ahighlight%283%29%29+

Replacing "b" with "13" p+=+%28-highlight%28b%29+%2B-+sqrt%28+highlight%28b%29%5E2-4%2A3%2Ac+%29%29%2F%282%2A3%29+ becomes p+=+%28-highlight%2813%29+%2B-+sqrt%28+highlight%2813%29%5E2-4%2A3%2Ac+%29%29%2F%282%2A3%29+

Replacing "c" with "12" p+=+%28-13+%2B-+sqrt%28+13%5E2-4%2A3%2Ahighlight%28c%29+%29%29%2F%282%2A3%29+ becomes p+=+%28-13+%2B-+sqrt%28+13%5E2-4%2A3%2Ahighlight%2812%29+%29%29%2F%282%2A3%29+

Mr. Ichudov's Quadratic Solver explains the remaining process better than I:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ap%5E2%2Bbp%2Bc=0 (in our case 3p%5E2%2B13p%2B12+=+0) has the following solutons:

p%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2813%29%5E2-4%2A3%2A12=25.

Discriminant d=25 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-13%2B-sqrt%28+25+%29%29%2F2%5Ca.

p%5B1%5D+=+%28-%2813%29%2Bsqrt%28+25+%29%29%2F2%5C3+=+-1.33333333333333
p%5B2%5D+=+%28-%2813%29-sqrt%28+25+%29%29%2F2%5C3+=+-3

Quadratic expression 3p%5E2%2B13p%2B12 can be factored:
3p%5E2%2B13p%2B12+=+3%28p--1.33333333333333%29%2A%28p--3%29
Again, the answer is: -1.33333333333333, -3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B13%2Ax%2B12+%29

So your solutions are: p=-1.33333333333333 and p=-3. and bob's your uncle!