SOLUTION: a. {{{ 12x - 5x^2 }}} What is the maximum value of this expression? b. {{{ y = 4x^2 + x - 2 }}} For which values of y will x be positive? Please show working, thanks in

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: a. {{{ 12x - 5x^2 }}} What is the maximum value of this expression? b. {{{ y = 4x^2 + x - 2 }}} For which values of y will x be positive? Please show working, thanks in      Log On


   

Question 985669: a.
+12x+-+5x%5E2+
What is the maximum value of this expression?
b.
+y+=+4x%5E2+%2B+x+-+2+
For which values of y will x be positive?
Please show working, thanks in advance
Found 2 solutions by macston, Alan3354:
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
a.
y=12x-5x%5E2+
Maximum occurs when first derivative=0
dy%2Fdx=12-10x Set=0
0=12-10x
10x=12
x=12%2F10=6%2F5 Maximum value when x=6/5.
.
y=12x-5x%5E2
y=12%286%2F5%29-5%286%2F5%29%5E2
y=72%2F5-5%2836%2F25%29
y=360%2F25-180%2F25=180%2F25=7.2
ANSWER: Maximum value is 7.2.
.
Please limit to one question per post.







Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
a.
+12x+-+5x%5E2+
What is the maximum value of this expression?
Without calculus:
f(x) = -5x^2 + 12x
The max is the vertex at x = -b/2a = -12/-10 = 1.2
f(1.2) = 14.4 - 7.2
= 7.2
======================
b.
+y+=+4x%5E2+%2B+x+-+2+
For which values of y will x be positive?
Find where y = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B1x%2B-2+=+0) has the following solutons:

x%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=%281%29%5E2-4%2A4%2A-2=33.

Discriminant d=33 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+33+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+33+%29%29%2F2%5C4+=+0.593070330817254
x%5B2%5D+=+%28-%281%29-sqrt%28+33+%29%29%2F2%5C4+=+-0.843070330817254

Quadratic expression 4x%5E2%2B1x%2B-2 can be factored:
4x%5E2%2B1x%2B-2+=+%28x-0.593070330817254%29%2A%28x--0.843070330817254%29
Again, the answer is: 0.593070330817254, -0.843070330817254. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B1%2Ax%2B-2+%29

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x = 0 at those 2 points, x1 & x2
y is positive for x < x1 and x > x2