SOLUTION: 24x²-2x=15 solve the equation by factoring and quadratic formula

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Question 102791: 24x²-2x=15 solve the equation by factoring and quadratic formula
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"24x²-2x=15 solve the equation by factoring"

24x%5E2-2x=15 Start with the given equation

24x%5E2-2x-15=0 Subtract 15 from both sides


%286x-5%29%284x%2B3%29=0 Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:

6x-5=0 or 4x%2B3=0

x=5%2F6 or x=-3%2F4 Now solve for x in each case


So our solutions are

x=5%2F6 or x=-3%2F4


Notice if we graph y=24x%5E2-2x-15 we get

+graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C+24x%5E2-2x-15%29+

and we can see that the graph has roots at x=5%2F6 and x=-3%2F4, so this verifies our answer.





"4x²-2x=15 solve the equation by the quadratic formula"


24x%5E2-2x=15 Start with the given equation

24x%5E2-2x-15=0 Subtract 15 from both sides


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


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

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



So lets solve 24%2Ax%5E2-2%2Ax-15=0 ( notice a=24, b=-2, and c=-15)




x+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A24%2A-15+%29%29%2F%282%2A24%29 Plug in a=24, b=-2, and c=-15



x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A24%2A-15+%29%29%2F%282%2A24%29 Negate -2 to get 2



x+=+%282+%2B-+sqrt%28+4-4%2A24%2A-15+%29%29%2F%282%2A24%29 Square -2 to get 4 (note: remember when you square -2, you must square the negative as well. This is because %28-2%29%5E2=-2%2A-2=4.)



x+=+%282+%2B-+sqrt%28+4%2B1440+%29%29%2F%282%2A24%29 Multiply -4%2A-15%2A24 to get 1440



x+=+%282+%2B-+sqrt%28+1444+%29%29%2F%282%2A24%29 Combine like terms in the radicand (everything under the square root)



x+=+%282+%2B-+38%29%2F%282%2A24%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%282+%2B-+38%29%2F48 Multiply 2 and 24 to get 48

So now the expression breaks down into two parts

x+=+%282+%2B+38%29%2F48 or x+=+%282+-+38%29%2F48

Lets look at the first part:

x=%282+%2B+38%29%2F48

x=40%2F48 Add the terms in the numerator
x=5%2F6 Divide

So one answer is
x=5%2F6



Now lets look at the second part:

x=%282+-+38%29%2F48

x=-36%2F48 Subtract the terms in the numerator
x=-3%2F4 Divide

So another answer is
x=-3%2F4

So our solutions are:
x=5%2F6 or x=-3%2F4

Notice when we graph 24%2Ax%5E2-2%2Ax-15, we get:

+graph%28+500%2C+500%2C+-13%2C+15%2C+-13%2C+15%2C24%2Ax%5E2%2B-2%2Ax%2B-15%29+

and we can see that the roots are x=5%2F6 and x=-3%2F4. This verifies our answer