SOLUTION: I need help solving this equation: {{{9a=-a^2-18}}}

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Question 227736: I need help solving this equation: 9a=-a%5E2-18
Found 2 solutions by jim_thompson5910, drj:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
9a=-a%5E2-18 Start with the given equation.


0=-a%5E2-9a-18 Subtract 9a from both sides.


Notice that the quadratic -a%5E2-9a-18 is in the form of Aa%5E2%2BBa%2BC where A=-1, B=-9, and C=-18


Let's use the quadratic formula to solve for "a":


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


a+=+%28-%28-9%29+%2B-+sqrt%28+%28-9%29%5E2-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29 Plug in A=-1, B=-9, and C=-18


a+=+%289+%2B-+sqrt%28+%28-9%29%5E2-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29 Negate -9 to get 9.


a+=+%289+%2B-+sqrt%28+81-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29 Square -9 to get 81.


a+=+%289+%2B-+sqrt%28+81-72+%29%29%2F%282%28-1%29%29 Multiply 4%28-1%29%28-18%29 to get 72


a+=+%289+%2B-+sqrt%28+9+%29%29%2F%282%28-1%29%29 Subtract 72 from 81 to get 9


a+=+%289+%2B-+sqrt%28+9+%29%29%2F%28-2%29 Multiply 2 and -1 to get -2.


a+=+%289+%2B-+3%29%2F%28-2%29 Take the square root of 9 to get 3.


a+=+%289+%2B+3%29%2F%28-2%29 or a+=+%289+-+3%29%2F%28-2%29 Break up the expression.


a+=+%2812%29%2F%28-2%29 or a+=++%286%29%2F%28-2%29 Combine like terms.


a+=+-6 or a+=+-3 Simplify.


So the solutions are a+=+-6 or a+=+-3

Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
I need help solving this equation: 9a=-a%5E2-18

Step 1. Add a%5E2%2B18 to both sides of the equation.

9a%2Ba%5E2%2B18=-a%5E2-18%2Ba%5E2%2B18

a%5E2%2B9a%2B18=0 where we simplified and arranged in descending order.

Step 2. To solve, use the quadratic equation given as

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

where a=1, b=9 and c=18.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B9x%2B18+=+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=%289%29%5E2-4%2A1%2A18=9.

Discriminant d=9 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-9%2B-sqrt%28+9+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%289%29%2Bsqrt%28+9+%29%29%2F2%5C1+=+-3
x%5B2%5D+=+%28-%289%29-sqrt%28+9+%29%29%2F2%5C1+=+-6

Quadratic expression 1x%5E2%2B9x%2B18 can be factored:
1x%5E2%2B9x%2B18+=+1%28x--3%29%2A%28x--6%29
Again, the answer is: -3, -6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B9%2Ax%2B18+%29



Step 3. ANSWER: The solutions are -3 and -6.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J
http://www.FreedomUniversity.TV