SOLUTION: How would I solve this? Find all real solutions of each equation. Please help me solve this equation: {{{ 2 sqrt (x^2+3)= sqrt (-16x-3) }}}

Algebra ->  Equations -> SOLUTION: How would I solve this? Find all real solutions of each equation. Please help me solve this equation: {{{ 2 sqrt (x^2+3)= sqrt (-16x-3) }}}      Log On


   

Question 154730: How would I solve this?
Find all real solutions of each equation.
Please help me solve this equation: +2+sqrt+%28x%5E2%2B3%29=+sqrt+%28-16x-3%29+
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
+2+sqrt+%28x%5E2%2B3%29=+sqrt+%28-16x-3%29+
Square both sides
4*(x^2 + 3) = -16x - 3
4x^2 + 12 = -16x - 3
4x^2 + 16x + 15 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B16x%2B15+=+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=%2816%29%5E2-4%2A4%2A15=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-16%2B-sqrt%28+16+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2816%29%2Bsqrt%28+16+%29%29%2F2%5C4+=+-1.5
x%5B2%5D+=+%28-%2816%29-sqrt%28+16+%29%29%2F2%5C4+=+-2.5

Quadratic expression 4x%5E2%2B16x%2B15 can be factored:
4x%5E2%2B16x%2B15+=+%28x--1.5%29%2A%28x--2.5%29
Again, the answer is: -1.5, -2.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B16%2Ax%2B15+%29