SOLUTION: Solve by using the quadratic formula: 9z squared + 12z + 4 = 0

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Question 114678: Solve by using the quadratic formula:
9z squared + 12z + 4 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's use the quadratic formula to solve for z:


Starting with the general quadratic

az%5E2%2Bbz%2Bc=0

the general solution using the quadratic equation is:

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



So lets solve 9%2Az%5E2%2B12%2Az%2B4=0 ( notice a=9, b=12, and c=4)




z+=+%28-12+%2B-+sqrt%28+%2812%29%5E2-4%2A9%2A4+%29%29%2F%282%2A9%29 Plug in a=9, b=12, and c=4



z+=+%28-12+%2B-+sqrt%28+144-4%2A9%2A4+%29%29%2F%282%2A9%29 Square 12 to get 144



z+=+%28-12+%2B-+sqrt%28+144%2B-144+%29%29%2F%282%2A9%29 Multiply -4%2A4%2A9 to get -144



z+=+%28-12+%2B-+sqrt%28+0+%29%29%2F%282%2A9%29 Combine like terms in the radicand (everything under the square root)



z+=+%28-12+%2B-+0%29%2F%282%2A9%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



z+=+%28-12+%2B-+0%29%2F18 Multiply 2 and 9 to get 18

So now the expression breaks down into two parts

z+=+%28-12+%2B+0%29%2F18 or z+=+%28-12+-+0%29%2F18

Lets look at the first part:

x=%28-12+%2B+0%29%2F18

z=-12%2F18 Add the terms in the numerator
z=-2%2F3 Divide

So one answer is
z=-2%2F3



Now lets look at the second part:

x=%28-12+-+0%29%2F18

z=-12%2F18 Subtract the terms in the numerator
z=-2%2F3 Divide

So another answer is
z=-2%2F3

So our only solution is:
z=-2%2F3

Notice when we graph 9%2Ax%5E2%2B12%2Ax%2B4 (just replace z with x), we get:

+graph%28+500%2C+500%2C+-12%2C+8%2C+-12%2C+8%2C9%2Ax%5E2%2B12%2Ax%2B4%29+

and we can see that the roots is x=-2%2F3. This verifies our answer