What quadratic equation has the roots 6 ± 3i?

Question

What quadratic equation has the roots 6 ± 3i?

 

A.x² + 45 = 0
B.x² + 81 = 0

C.x² – 12x + 45 = 0
D.x² – 12x + 811 = 0

 

 

Answer 1

The answer is C.

 

You must use the quadratic formula to determine the roots of the equation. If you write out the quadratic formula for each of the four equations listed as possible solutions, you can quickly eliminate answer A and B. 

 

quadratic formula: [-b ±√(b²-4ac)] / 2a

 

So, for A and B, the b term is 0, (x²+0x+45 = 0, x²+0x+81 = 0)

 

For C and D, the b term is -12, a is 1, and c is the only difference between the two. So now, you just need to look at which of these give you -36 under the radical (-4ac = -36). And if you try both, you will find that only answer C works.

 

So if you write out the quadratic formula for equation C, yo get:

 

{-(-12)±√[(-12)²-4(1)(45)] }/ 2(1)

 

[ 12±√(-36) ] / 2

 

(12±6i)/2

 

6±3i

 

I hope the brackets and parenthesis are not confusing, I just want to emphasize the correct order of operations.

Answer 2

Hi Makayla,

 

 

Eliminate A and B right away because the square roots of -45 and -83 would not give you any constant terms.

 

Then (working on choice C, and using just the determinant b^2 - 4ac of the quadratic equation and not the whole

 

-b+/-square root of b^-4ac) subtract 4*45 from 144 and get -36. Then stop because even if you haven't checked to

 

see if the whole thing works out just the determinant tells you that D definitely is not an answer. 

 

I hope this helps

Answer 3

Hi Makayla

 

if the roots are 6+3i and 6-3i you can use these to get the factors as follows

 

x = 6+3i

 

x = 6-3i

 

so

 

(x - (6+3i))(x- (6-3i))=0

 

and you can multipy using the FOIL method

 

x² -x(6-3i) -x(6+3i) + 6² - 9i² =0

 

x² -6x +3xi -6x - 3xi + 36 -(-9) = 0

 

x² -12x +45 = 0

 

Hope this helps

 

x

Answer 4

If the roots are 6±3i, then the factors of the quadratic are:

 

(x - (6+3i))(x - (6-3i))

 

Now just multiply them out using the FOIL method:

 

x*x - x*(6-3i) - (6+3i)*x + (6-3i)(6+3i)

 

x² - 6x + 3xi - 6x - 3xi + 36 + 18i - 18i - (3i)²

 

x² - 12x + 36 + 9

 

x² - 12x + 45