SOLUTION: 3x^2-1=5x^2-3x-5

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Question 175637This question is from textbook
: 3x^2-1=5x^2-3x-5 This question is from textbook

Found 2 solutions by jim_thompson5910, Mathtut:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3x%5E2-1=5x%5E2-3x-5 Start with the given equation.


0=5x%5E2-3x-5-3x%5E2%2B1 Subtract 3x%5E2 from both sides. Add 1 to both sides.


0=2x%5E2-3x-4 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-3, and c=-4


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-%28-3%29+%2B-+sqrt%28+%28-3%29%5E2-4%282%29%28-4%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-3, and c=-4


x+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%282%29%28-4%29+%29%29%2F%282%282%29%29 Negate -3 to get 3.


x+=+%283+%2B-+sqrt%28+9-4%282%29%28-4%29+%29%29%2F%282%282%29%29 Square -3 to get 9.


x+=+%283+%2B-+sqrt%28+9--32+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-4%29 to get -32


x+=+%283+%2B-+sqrt%28+9%2B32+%29%29%2F%282%282%29%29 Rewrite sqrt%289--32%29 as sqrt%289%2B32%29


x+=+%283+%2B-+sqrt%28+41+%29%29%2F%282%282%29%29 Add 9 to 32 to get 41


x+=+%283+%2B-+sqrt%28+41+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%283%2Bsqrt%2841%29%29%2F%284%29 or x+=+%283-sqrt%2841%29%29%2F%284%29 Break up the expression.


So the answers are x+=+%283%2Bsqrt%2841%29%29%2F%284%29 or x+=+%283-sqrt%2841%29%29%2F%284%29


which approximate to x=2.351 or x=-0.851

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
3x%5E2-1=5x%5E2-3x-5given
:
2x%5E2-3x-4=0combining like terms on one side
:
:
solutionsystem%282.35%2C-.85%29
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-3x%2B-4+=+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=%28-3%29%5E2-4%2A2%2A-4=41.

Discriminant d=41 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+41+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+41+%29%29%2F2%5C2+=+2.35078105935821
x%5B2%5D+=+%28-%28-3%29-sqrt%28+41+%29%29%2F2%5C2+=+-0.850781059358212

Quadratic expression 2x%5E2%2B-3x%2B-4 can be factored:
2x%5E2%2B-3x%2B-4+=+%28x-2.35078105935821%29%2A%28x--0.850781059358212%29
Again, the answer is: 2.35078105935821, -0.850781059358212. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-3%2Ax%2B-4+%29