ax2 + bx + c where a ( ≠ 0 ), b, c are constants which can take real number values,is called a Quadratic Polynomial or Quadratic Expression in x.
If the Quadratic polynomial or Expression is equated to zero, it is called a Quadratic Equation.
The two methods to solve Quadratic Equations are by (i) Factorization of Quadratic Polynomials to get two linear factors, discussed in Algebra Factoring.
and by (ii) Formula known as
Quadratic Formula. We make use of that Formula here. We also need the knowledge of Factorization of Quadratic Expressions, here.
Here, we solve the Quadratic Expressions with the signs 'greater than' (>) or 'less than' (<), or 'greater than or equal to' (≥) or 'less than or equal to' (≤).
These are called Quadratic Inequalities.
Solving Quadratic Inequalities :
Formula 1 of Quadratic Inequalities
(x - α)(x - β) > 0 ⇒ x ∉ (α, β) or x ∈ (-∞,α) ∪ (β, ∞) or α > x > β i.e. x doenot lie between α and β
Proof: We have α < β The product of (x - α) and (x - β) is positive only whenboth are positive or both are negative.We can see, If x > β, both are positive. similarly, If x < α, both are negative. ∴ (x - α)(x - β) > 0 ⇒ x doenot lie between α and β (proved.)
Formula 2 of Quadratic Inequalities
(x - α)(x - β) < 0 ⇒ x ∈ (α, β) or α < x < β i.e. x lies between α and β
Proof: The product of (x - α) and (x - β) is negative only whenone is positive and the other is negative. This is possible only when x lies between α and β ∴ (x - α)(x - β) < 0 ⇒ x lies between α and β (proved.)
Get The Best Grades With the Least Amount of Effort
Here is a collection of proven tips, tools and techniques to turn you into a super-achiever - even if you've never thought of yourself as a "gifted" student.
The secrets will help you absorb, digest and remember large chunks of information quickly and easily so you get the best grades with the least amount of effort.
If you apply what you read from the above collection, you can achieve best grades without giving up your fun, such as TV, surfing the net, playing video games or going out with friends!
There is an exclusive, Parent Information Page provides YOU with detailed reports of your child’s progress so you can monitor your child’s success and give them encouragement. These Reports include
Time spent using the program
Personalized remediation curriculum designed for your child
Details the areas of weakness where your child needs additional help
Provides the REASONS WHY your child missed a concept
List of modules accessed and amount of time spent in each module
Creates reports that can be printed and used to discuss issues with your child’s teachers
These reports are created and stored in a secure section of the program, available exclusively to you, the parent. The section is accessed by a password that YOU create and use. No unauthorized users can access this information.
If x2 + (b⁄a)x + (c⁄a) = 0, has no real roots ( i.e. has imaginary roots),then x2 + (b⁄a)x + (c⁄a) < 0 ⇒ x ∉ R (the real number set) or x ∉ (-∞, ∞) i.e. x can not any take any real value.
Proof : Based on the proof of previous Formula, x2 + (b⁄a)x + (c⁄a) can not be negativefor any real value of x. Hence x2 + (b⁄a)x + (c⁄a) < 0 has no solution in R. (Proved.)
Formula 6 of Quadratic Inequalities is thus proved.
Method of solving Quadratic Inequalities :
Method of solving a Quadratic Inequation such as, for a ≠ 0, ax2 + bx + c > 0 or ax2 + bx + c < 0
STEP 1 : Divide both sides of the inequation by a. Then the L.H.S. of the inequation becomes x2 + (b⁄a)x + (c⁄a) The inequality sign may remain the same (if a is positive), or may get reversed (if a is negative). The R.H.S.of the inequation will remain 0.
STEP 2 : Consider the roots of the quadratic equation x2 + (b⁄a)x + (c⁄a) = 0.
Case (i) : Let the roots be real and unequal.( i.e. the Discriminant, Δ > 0). Let the roots be α and β, such that α < β, Then, express x2 + (b⁄a)x + (c⁄a) as (x - α)(x - β)
Case (ii) : Let the roots be real and equal. ( i.e. the Discriminant, Δ = 0). Let the equal roots be αThen, express x2 + (b⁄a)x + (c⁄a) as (x - α)2
Case (iii) : Let the roots be imaginary.( i.e. the Discriminant, Δ < 0).
For the meaning of Discriminant, Δ and the nature of the roots of a quadratic equation, you may refer to
STEP 3 : Make use of formulas 1 and 2 for case (i) to solve the inequation. Make use of formulas 3 and 4 for case (ii) to solve the inequation. Make use of formulas 5 and 6 for case (iii) to solve the inequation.
NOTE : To solve ax2 + bx + c ≥ 0 or ax2 + bx + c ≤ 0, Make the slight modification to the solution, considering the equality symbol.
The three steps and this slight modification for the equality symbol, will be clear by the following sets of solved examples on Quadratic inequalities.
Progressive Learning of Math : Quadratic inequalities
Recently, I have found a series of math curricula (Both Hard Copy and Digital Copy) developed by a Lady Teacher who taught everyone from Pre-K students to doctoral students and who is a Ph.D. in Mathematics Education.
This series is very different and advantageous over many of the traditional books available. These give students tools that other books do not. Other books just give practice. These teach students “tricks” and new ways to think.
These build a student’s new knowledge of concepts from their existing knowledge. These provide many pages of practice that gradually increases in difficulty and provide constant review.
These also provide teachers and parents with lessons on how to work with the child on the concepts.
The series is low to reasonably priced and include