SOLUTION: How do a use the quadratic formula to solve this partial equation.. (4x+1)^2 = 3x+4

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Question 245847: How do a use the quadratic formula to solve this partial equation..
(4x+1)^2 = 3x+4
Found 3 solutions by nerdybill, dabanfield, jsmallt9:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
You would:
- expand the left side
- move all terms to the left
- then apply the quadratic
.
(4x+1)^2 = 3x+4
(4x+1)(4x+1) = 3x+4
16x^2+4x+4x+1 = 3x+4
16x^2+8x+1 = 3x+4
16x^2+5x+1 = 4
16x^2+5x-3 = 0
.
Applying the quadratic yields:
x = {0.3041, -0.6166}
.
Details of quadratic to follow:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 16x%5E2%2B5x%2B-3+=+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=%285%29%5E2-4%2A16%2A-3=217.

Discriminant d=217 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+217+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+217+%29%29%2F2%5C16+=+0.304091245708007
x%5B2%5D+=+%28-%285%29-sqrt%28+217+%29%29%2F2%5C16+=+-0.616591245708007

Quadratic expression 16x%5E2%2B5x%2B-3 can be factored:
16x%5E2%2B5x%2B-3+=+16%28x-0.304091245708007%29%2A%28x--0.616591245708007%29
Again, the answer is: 0.304091245708007, -0.616591245708007. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+16%2Ax%5E2%2B5%2Ax%2B-3+%29

Answer by dabanfield(803) About Me  (Show Source):
You can put this solution on YOUR website!
Expand (4x+1)^2 = 16x^2 + 8x + 1
Substutie the above in the original equation:
16x^2 + 8x + 1 = 3x + 4
Collecting terms:
16x^2 + 8x - 3x + 1 - 4 = 0
16x^2 + 5x - 3 = 0
It only remains to substitute a=16, b=8 and c = -3 in the quadratic formula to get the solutions.

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%284x%2B1%29%5E2+=+3x%2B4
To use the quadratic formula, x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F2a, your equation needs to be in ax%5E2+%2B+bx+%2B+c+=+0 form. So getting the equation into the proper form is where we start.
Multiply out the left side:
16x%5E2+%2B+8x+%2B+1+=+3x+%2B+4
Make the right side zero by subtracting 3x and 4 from each side:
16x%5E2+%2B+5x+-+3+=+0
Now we have the proper form and we can use the quadratic formula with "a" = 16, "b" = 5 and "c" = -3:
x+=+%28-%285%29+%2B-+sqrt%28%285%29%5E2+-+4%2816%29%28-3%29%29%29%2F2%2816%29
Now we simplify:
x+=+%28-%285%29+%2B-+sqrt%2825+-+4%2816%29%28-3%29%29%29%2F2%2816%29
x+=+%28-5+%2B-+sqrt%2825+%2B+192%29%29%2F32
x+=+%28-5+%2B-+sqrt%28217%29%29%2F32
So
x+=+%28-5+%2B+sqrt%28217%29%29%2F32 or x+=+%28-5+-+sqrt%28217%29%29%2F32