SOLUTION: What is the correct set up for this problem and how do I arrive at answer of a = -1/3. 5a + 1/4a + 6 = 3/4 I figured the LCD to be 2(2a + 3) 2(2a + 3)* 5a + 1/2(2a + 3) =

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: What is the correct set up for this problem and how do I arrive at answer of a = -1/3. 5a + 1/4a + 6 = 3/4 I figured the LCD to be 2(2a + 3) 2(2a + 3)* 5a + 1/2(2a + 3) =       Log On


   

Question 402215: What is the correct set up for this problem and how do I arrive at answer of a = -1/3.
5a + 1/4a + 6 = 3/4
I figured the LCD to be 2(2a + 3)
2(2a + 3)* 5a + 1/2(2a + 3) = 2(2a + 3)*3/4
= 5a + 1 = 2a + 3 * 3/2 (This is where I am having problem in solving this problem.
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
5a + 1/(4a+6) = 3/4
5a + 1/(2(2a+3)) = 3/4
multiply both sides by 4(2a+3)
5a(4(2a+3)) + 2 = 3(2a+3)
5a(8a+12) + 2 = 3(2a+3)
40a^2+60a + 2 = 6a+9
40a^2+54a + 2 = 9
40a^2+54a-7 = 0
apply the "quadratic formula" to get:
a = {0.119, -1.469}
.
Details of quadratic:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aa%5E2%2Bba%2Bc=0 (in our case 40a%5E2%2B54a%2B-7+=+0) has the following solutons:

a%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=%2854%29%5E2-4%2A40%2A-7=4036.

Discriminant d=4036 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-54%2B-sqrt%28+4036+%29%29%2F2%5Ca.

a%5B1%5D+=+%28-%2854%29%2Bsqrt%28+4036+%29%29%2F2%5C40+=+0.11911900871343
a%5B2%5D+=+%28-%2854%29-sqrt%28+4036+%29%29%2F2%5C40+=+-1.46911900871343

Quadratic expression 40a%5E2%2B54a%2B-7 can be factored:
40a%5E2%2B54a%2B-7+=+40%28a-0.11911900871343%29%2A%28a--1.46911900871343%29
Again, the answer is: 0.11911900871343, -1.46911900871343. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+40%2Ax%5E2%2B54%2Ax%2B-7+%29