SOLUTION: solve 4w^2=10w-4

Algebra ->  Equations -> SOLUTION: solve 4w^2=10w-4      Log On


   

Question 96217: solve
4w^2=10w-4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
4w%5E2=10w-4


4w%5E2-10w%2B4=0 Get all terms to one side



Let's use the quadratic formula to solve for w:


Starting with the general quadratic

aw%5E2%2Bbw%2Bc=0

the general solution using the quadratic equation is:

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

So lets solve 4%2Aw%5E2-10%2Aw%2B4=0 ( notice a=4, b=-10, and c=4)

w+=+%28--10+%2B-+sqrt%28+%28-10%29%5E2-4%2A4%2A4+%29%29%2F%282%2A4%29 Plug in a=4, b=-10, and c=4



w+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%2A4%2A4+%29%29%2F%282%2A4%29 Negate -10 to get 10



w+=+%2810+%2B-+sqrt%28+100-4%2A4%2A4+%29%29%2F%282%2A4%29 Square -10 to get 100 (note: remember when you square -10, you must square the negative as well. This is because %28-10%29%5E2=-10%2A-10=100.)



w+=+%2810+%2B-+sqrt%28+100%2B-64+%29%29%2F%282%2A4%29 Multiply -4%2A4%2A4 to get -64



w+=+%2810+%2B-+sqrt%28+36+%29%29%2F%282%2A4%29 Combine like terms in the radicand (everything under the square root)



w+=+%2810+%2B-+6%29%2F%282%2A4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



w+=+%2810+%2B-+6%29%2F8 Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

w+=+%2810+%2B+6%29%2F8 or w+=+%2810+-+6%29%2F8

Lets look at the first part:

x=%2810+%2B+6%29%2F8

w=16%2F8 Add the terms in the numerator
w=2 Divide

So one answer is
w=2



Now lets look at the second part:

x=%2810+-+6%29%2F8

w=4%2F8 Subtract the terms in the numerator
w=1%2F2 Divide

So another answer is
w=1%2F2

So our solutions are:
w=2 or w=1%2F2

Notice when we graph 4%2Ax%5E2-10%2Ax%2B4 (just replace w with x), we get:

+graph%28+500%2C+500%2C+-9%2C+12%2C+-9%2C+12%2C4%2Ax%5E2%2B-10%2Ax%2B4%29+

and we can see that the roots are w=2 and w=1%2F2. This verifies our answer