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# Hello!

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Can someone check my work/ I'm keep on getting $$\left[-\frac{3}{2},\:\frac{2}{5}\right]$$ as my answer.

Question: Find the domain of the real-valued function $$f(x)=\sqrt{-10x^2-11x+6}.$$Give the endpoints in your answer as common fractions, not mixed numbers or decimals.

Nov 28, 2018

#1
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We need to have  [ since the result inside the radical can't be negative ]

-10x^2 - 11x + 6  ≥  0     (1)     multiply both sides by  -1,  reverse the inequality sign

10x^2 + 11x - 6 ≤ 0      set to 0 and factor

10x^2 + 11x - 6  =  0

(5x -2) ( 2x + 3) = 0

Set each factor to 0 and solve for x

x =  2/5        or    x = -3/2

The intervals that make (1)  true come from either

( -inf, -3/2)    or    [ -3/2, 2/5]     or  (2/5, inf)

Test a point in the middle interval   - I'll pick 0 -  and test it in (1)

If  0, makes it true, then this is the correct interval....so...

-10(0)^2  - 11(0)+ 6  ≥  0        is true

So...you are correct Ant !!!.....the answer is    [ -3/2, 2/5 ]

Nov 28, 2018
edited by CPhill  Nov 29, 2018
#2
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Thank you, CPhill! I understand better now!

ant101  Nov 29, 2018
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Thanks, Ant !!!

CPhill  Nov 29, 2018