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2 solutions
this is exactly how i solved this
In fact extending the above argument it is easy to see that f ( 1 ) < 0 and f ( − 1 ) < 0 are necessary and sufficient for the roots of the quadratic to be in ( − ∞ , − 1 ) and ( 1 , ∞ ) . Hence, m 2 − 8 < 0 and m 2 + 2 m − 6 < 0 , which implies that m ∈ ( − 2 ( 2 ) , 2 ) . Thus the only integer values that are acceptable are { − 2 , − 1 , 0 , 1 } giving the answer as 4 and not 5 .
Using the discriminant for the polynomial: (m + 1)^2 -4(m^2 + m - 8) >= 0 to have real solutions. This gives -11/3 <= m <= 3. This gives 7 possible integral m solutions. All integers in that range, except for 3 and -3 since they give roots of multiplicity 2 for m = 3 and solutions both less than 1 for m = -3. Hence, 5 solutions.
Pal you may have used latex your answer would look more beautiful. I am not a moderator thus I can not latexify it.
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Sorry... I have no knowledge of using LaTex...
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Then use "Daum Equation Editor" in Chrome Browser.
Using the discriminant for the polynomial:
(
m
+
1
)
2
−
4
(
m
2
+
m
−
8
)
>
=
0
to have real solutions. This gives
3
−
1
1
<
=
m
<
=
3
. This gives 7 possible integral m solutions. All integers in that range, except for 3 and -3 since they give roots of multiplicity 2 for m = 3 and solutions both less than 1 for m = -3. Hence, 5 solutions.
As desired by Parth Lohomi .
Let the roots of the equation be a , b where a < b
Now, according to the problem a < 1 and b > 1
And, we know that, since the leading coefficient is positive, the graph of the given quadratic must be a parabola (concave upwards).
Now, from this, we arrive at the conclusion that f ( 1 ) < 0
For instance, the graph will be something like -
(Thus, the graph clearly suggests that since one root is grater than 1 and other being less than 1, f ( 1 ) must necessarily be negative)
Thus, f ( 1 ) = 1 − ( m + 1 ) + m 2 + m − 8 < 0 m 2 − 8 < 0 ( m − 2 2 ) ( m + 2 2 ) < 0 ⇒ − 2 2 < m < 2 2
Therefore, the only possible integer values of m are ( − 2 , − 1 , 0 , 1 , 2 ) which gives total 5 integral values for m
(Note - The original graph may be completely different. I've used this graph just to demonstrate about the behaviour of the given quadratic function)