An equation that does not have an equal to sign and has the highest degree 2 is called quadratic inequality. These equations will have signs such as less than, greater than, less than or equal, and greater than or equal. Solving these inequalities is the same as solving second-degree polynomial equations. As far as the JEE exam is concerned, this is an important topic. In this article, we will discuss properties of inequalities, properties of intervals, etc.
Consider an equation x2-3x-10 <0. This is an example of a quadratic inequality. The following are types of quadratic inequalities.
- Inequalities involving non-repeating linear factors
- Inequalities involving repeating linear factors
- Inequalities expressed in rational form.
- Double inequality
- Inequalities involving bi-quadrate expression
Properties of inequality
- We can add or subtract any number from both sides of an inequality.
- The sign of inequality does not change if we shift the terms one side to the other side of the inequality.
- If we multiply both sides of the inequality by a non zero positive number, then the sign of inequality remains unchanged.
- If we multiply both sides of the inequality by a non-zero negative number then the sign of the inequality changes.
- If the sign of an expression is not known in inequality, then it cannot be cross multiplied. Also without knowing the sign of expression, division is not possible.
Properties of inequality intervals
- Closed Interval: It is denoted by x ϵ [a, b]. i.e. a ≤ x ≤ b. This means the values of x that lie between a & b and is also equal to a and b.
- Open closed Interval: It is denoted by x ϵ (a, b]. i.e. a < x ≤ b. In this interval, the values of x lie between a and b, equal to b, but not equal to a.
- Closed-open Interval: It is denoted by c ϵ [a, b). i.e. a ≤ x <b. In this interval, the values of x lie between a and b, equal to a, but not equal to b.
An equation of the form ax2+bx+c = 0, is known as a quadratic equation. a,b,c are real numbers and a should not be equal to zero. The highest power of this equation is 2. The values of x satisfying this equation are known as the roots of the equation. A second-degree polynomial equation will have two roots. The roots may be either real or imaginary. Many physical and mathematical problems are in the form of second-degree polynomial equations. In mathematics, the solutions of these equations are of greater importance.
The general form is given by ax2+bx+c = 0. The following are some examples of quadratic equations.
- x2+6x+9 = 0
- x2-x-6 = 0
- 2x2-3x+9 = 0
The solution is given by the equation [-b±√(b2-4ac])/2a. This is known as the quadratic formula.
b2-4ac is called the discriminant of the quadratic formula. It is denoted by D.
Nature of roots
- If D > 0, then the roots will be real and distinct.
- If D = 0, then the roots will be equal and real.
- If D < 0, then the roots will be imaginary.