SOLUTION: Solve equation x^2+2x+5=0 value of k that completes the square x^2+10x+k

Algebra ->  Equations -> SOLUTION: Solve equation x^2+2x+5=0 value of k that completes the square x^2+10x+k       Log On


   

Question 77762This question is from textbook
: Solve equation
x^2+2x+5=0
value of k that completes the square
x^2+10x+k
 This question is from textbook

Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To solve for x, we need to use the quadratic formula:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B2x%2B5+=+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=%282%29%5E2-4%2A1%2A5=-16.

The discriminant -16 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -16 is + or - sqrt%28+16%29+=+4.

The solution is x%5B12%5D+=+%28-2%2B-+i%2Asqrt%28+-16+%29%29%2F2%5C1+=++%28-2%2B-+i%2A4%29%2F2%5C1+

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B5+%29


So our answer is
x=-1%2B2i or x=-1-2i
The value of k that completes the square will be equal to the square of half of 10, which looks like this
k=%2810%2F2%29%5E2
k=5%5E2
k=25

So our polynomial becomes
x%5E2%2B10x%2B25
Notice how it can be factored to %28x%2B5%29%28x%2B5%29=%28x%2B5%29%5E2 which is a perfect square

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Solve equation
x^2+2x+5=0
Use the quadratic formula to get:
x=[-2+-sqrt(4-4*5)]/2
x=[-2+-4i]/2
x= -1+2i ; or x = -1-2i
=======================
value of k that completes the square
x^2+10x+k
x^2+10x+(10/2)^2
=x^2+10x+25
k=25
==============
Cheers,
Stan H.