Property 1
\[ f(x)=ax^2+bx+c \] \[ \Delta=b^2-4ac>0 \]| x | −∞ | x₁ | x₂ | +∞ | |
|---|---|---|---|---|---|
| f(x) | + | ○ | − | ○ | + |
| x | −∞ | x₁ | x₂ | +∞ | |
|---|---|---|---|---|---|
| f(x) | − | ○ | + | ○ | − |
In quadratic equations the roots are found by factorization. If “a” is a positive factor then the expression between the roots is negative, in the other intervals it will be positive. If “a” is a negative factor then between roots it is positive and in other intervals is negative.
Example
\(x^2-x-12>0\)
Answer
\((x-4)(x+3)>0\)
| x | −∞ | −3 | 4 | +∞ | |
|---|---|---|---|---|---|
| f(x) | + | ○ | − | ○ | + |
\((−\infty,-3)\cup(4,+\infty)\)
\(R=[-3,4]\)
Example
\((3-x)(x+7)\ge0\)
Answer
| x | −∞ | −7 | 3 | +∞ | |
|---|---|---|---|---|---|
| f(x) | − | ○ | + | ○ | − |
\([-7,3]\)
1.
\(x(x+6)\le0\)
\(\Rightarrow S.S.=?\)
2.
\(x^2-3x\ge0\)
\(\Rightarrow S.S.=?\)
3.
\(x^2-4x+3<0\)
\(\Rightarrow S.S.=?\)
4.
\(x^2-36<0\)
\(\Rightarrow S.S.=?\)
5.
\(2x^2-2x-3
\(\Rightarrow S.S.=?\)
6.
\(2x^2-11x+12>0\)
\(\Rightarrow S.S.=?\)
7.
\(x\in\mathbb{Z}^+\)
\(8x-12\ge x^2\)
\(\Rightarrow\sum x=?\)