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Let a, b, c real. If ax^(2)+bx+c=0 has t...

Let a, b, c real. If `ax^(2)+bx+c=0` has two real roots `alpha and beta,` where `alpha lt -1and betagt1,` then show that `1+c/a+|(b)/(a)|lt0`

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Let `f(x)= x^(2)+ (b)/(a) x + (c)/(a)`.
According to the questions, we have the following graph.

From the graph
`" "f(-1) lt 0 and f(1) lt 0`
`rArr" "1 + (c)/(a)- (b)/(a)lt 0 and 1+ (c)/(a)+ (b)/(a) lt 0`
`rArr" "1+ (c)/(a)+ |(b)/(a)| lt 0`
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CENGAGE-GRAPHS OF ELEMENTARY FUNCTIONS -Exercise
  1. Let a, b, c real. If ax^(2)+bx+c=0 has two real roots alpha and beta, ...

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  2. Draw the graph of y= (1)/((1//x)).

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  3. (a) Draw the graph of f(x) = ={{:(1",",, |x| ge 1), ((1)/(n^(2)) ",",,...

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  4. Sketch the regions which points satisfy |x+y| ge 2.

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  5. Sketch the region satisfying |x| lt |y|.

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  6. Let f:R→R:f(x)=(x+1) and g:R→R:g(x)=(x^2−2). Write down the formulae f...

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  7. Draw the graph of y= (x-1)/(x-2).

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  8. The following figure shows the graph of f(x) =ax^(2)+bx +c, then find ...

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  9. The entire graph of the equation y=x^2+k x-x+9 in strictly above the x...

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  10. If x^2+2a x+a<0AAx in [1,3], the find the values of adot

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  11. Draw the graph of f(x) = x|x|.

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  12. Draw the graph of the function: Solve |(x^2)/(x-1)|lt=1 using the grap...

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  13. Draw the graph of y = |x^(2) - 2x|-x.

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  14. Draw the graph of y =2^(x)"," x^(2)-2x le 0

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  15. Find the roots of the equation by factorization: 2x^2-x-1

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  16. Divide 16(x^2yz + xy^2z+xyz^2) by 4xyz

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  17. Find the set of real value(s) of a for which the equation |2x+3|+2x-3|...

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  18. Draw the graph of y=|x|.

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  19. Draw the graph of y=1/(log(e)x)

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  20. Find the number of real solutions to the equation log(0.5)x=|x|.

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  21. Draw the graph of f(x)= x+ [x], where [*] denotes the greatest integer...

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