Many inequalities lead to finding the sign of a quadratic expression. let us discuss this problem here. Consider the quadratic function

We know that

**1**- if (double root case), then we have
In this case, the function has the sign of the coefficient

*a*.**a<0****a>0** **2**- If (two distinct real roots case). In this case, we have
where and are the two roots with . Since is always positive when and , and always negative when , we get

- has same sign as the coefficient
*a*when and ; - has opposite sign as the coefficient
*a*when .

**a<0****a>0** - has same sign as the coefficient
**3**- If (complex roots case), then has a constant sign same as the coefficient
*a*.**a<0****a>0**

**Example:** Solve the inequality

**Solution.** First let us find the root of the quadratic equation . The quadratic formula gives

which yields *x*= -1 or *x*=2. Therefore, the expression
is negative or equal to 0 when .

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