SOLUTION: Hi. Can I have some help finishing this expression? Find all solutions of the equation and express them in the form a + bi. x^2 + 7x + 12 = 0

Algebra ->  Equations -> SOLUTION: Hi. Can I have some help finishing this expression? Find all solutions of the equation and express them in the form a + bi. x^2 + 7x + 12 = 0       Log On


   

Question 120811: Hi. Can I have some help finishing this expression?
Find all solutions of the equation and express them in the form a + bi.
x^2 + 7x + 12 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve x%5E2%2B7%2Ax%2B12=0 ( notice a=1, b=7, and c=12)




x+=+%28-7+%2B-+sqrt%28+%287%29%5E2-4%2A1%2A12+%29%29%2F%282%2A1%29 Plug in a=1, b=7, and c=12



x+=+%28-7+%2B-+sqrt%28+49-4%2A1%2A12+%29%29%2F%282%2A1%29 Square 7 to get 49



x+=+%28-7+%2B-+sqrt%28+49%2B-48+%29%29%2F%282%2A1%29 Multiply -4%2A12%2A1 to get -48



x+=+%28-7+%2B-+sqrt%28+1+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-7+%2B-+1%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-7+%2B-+1%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%28-7+%2B+1%29%2F2 or x+=+%28-7+-+1%29%2F2

Lets look at the first part:

x=%28-7+%2B+1%29%2F2

x=-6%2F2 Add the terms in the numerator
x=-3 Divide

So one answer is
x=-3



Now lets look at the second part:

x=%28-7+-+1%29%2F2

x=-8%2F2 Subtract the terms in the numerator
x=-4 Divide

So another answer is
x=-4

So our solutions are:
x=-3 or x=-4

which look like this in a%2Bbi form

x=-3%2B0i or x=-4%2B0i




Notice when we graph x%5E2%2B7%2Ax%2B12, we get:

+graph%28+500%2C+500%2C+-14%2C+7%2C+-14%2C+7%2C1%2Ax%5E2%2B7%2Ax%2B12%29+

and we can see that the roots are x=-3 and x=-4. This verifies our answer