SOLUTION: 17. Express the roots of 9x^2 + 40 = 36x in simplest (a + bi) form. Thanks happy new years

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Question 174716This question is from textbook Amsco's Preparing for the Regents Examination Mathematics B
: 17.
Express the roots of 9x^2 + 40 = 36x in simplest (a + bi) form.
Thanks happy new years This question is from textbook Amsco's Preparing for the Regents Examination Mathematics B

Found 2 solutions by Mathtut, jim_thompson5910:
Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
9x%5E2-36%2B40=0
rootssystem%282%2B%282%2F3%29i%2C2-%282%2F3%29i%29
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B-36x%2B40+=+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=%28-36%29%5E2-4%2A9%2A40=-144.

The discriminant -144 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 -144 is + or - sqrt%28+144%29+=+12.

The solution is x%5B12%5D+=+%28--36%2B-i%2Asqrt%28+-144+%29%29%2F2%5C9+=++%28--36%2B-i%2A12%29%2F2%5C9+, or
Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-36%2Ax%2B40+%29

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

9x%5E2%2B40=36x Start with the given equation.


9x%5E2-36x%2B40=0 Subtract 36x from both sides.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=9, b=-36, and c=40


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-36%29+%2B-+sqrt%28+%28-36%29%5E2-4%289%29%2840%29+%29%29%2F%282%289%29%29 Plug in a=9, b=-36, and c=40


x+=+%2836+%2B-+sqrt%28+%28-36%29%5E2-4%289%29%2840%29+%29%29%2F%282%289%29%29 Negate -36 to get 36.


x+=+%2836+%2B-+sqrt%28+1296-4%289%29%2840%29+%29%29%2F%282%289%29%29 Square -36 to get 1296.


x+=+%2836+%2B-+sqrt%28+1296-1440+%29%29%2F%282%289%29%29 Multiply 4%289%29%2840%29 to get 1440


x+=+%2836+%2B-+sqrt%28+-144+%29%29%2F%282%289%29%29 Subtract 1440 from 1296 to get -144


x+=+%2836+%2B-+sqrt%28+-144+%29%29%2F%2818%29 Multiply 2 and 9 to get 18.


x+=+%2836+%2B-+12%2Ai%29%2F%2818%29 Take the square root of -144 to get 12%2Ai.


x+=+%2836+%2B+12%2Ai%29%2F%2818%29 or x+=+%2836+-+12%2Ai%29%2F%2818%29 Break up the expression.


x+=+%2836%29%2F%2818%29+%2B+%2812%2Ai%29%2F%2818%29 or x+=++%2836%29%2F%2818%29+-+%2812%2Ai%29%2F%2818%29 Break up the fraction for each case.


x+=+2%2B%282%2F3%29%2Ai or x+=+2-%282%2F3%29%2Ai Reduce.


So the answers are x+=+2%2B%282%2F3%29%2Ai or x+=+2-%282%2F3%29%2Ai