Quadratic Equations - Sum and Product of Roots of Quadratic and Higher Polynomials, Discriminant, Maximum and Minimum Value, Graphical Representation Video

A general quadratic equation is represented by ax²+bx+c = 0 where a is not equal to zero and a,b,c are real numbers. The degree of this equation is two, so the maximum number of solution of this equation are two. If we know the quadratic equation then we can calculate the roots of that particular equation and if we know the roots of a particular quadratic equation then from those roots I can make that quadratic equation. Most of the questions of this topic revolve around this particular concept. If the roots are alpha and beta then the quadratic equation will be -

1. (x-alpha)(x-beta) = 0
2. x²-(alpha+beta)x+(alpha)(beta) = 0 whose general form is x²-(sum of roots)x+product of roots = 0

A general quadratic equation is represented by ax²+bx+c = 0 , then the formula of roots would be (-b plus minus under root discriminant)/2a.Discriminant of this quadratic equation becomes b² - 4ac and the sum of the roots becomes -b/a and the product of the roots of this quadratic equation becomes c/a.

When the discriminant of the quadratic equation greater than 0, then the roots will become unequal, real and rational if the discriminant is a perfect square.

When the discriminant of the quadratic equation is lesser than 0, then the roots will become complex (combination of real and imaginary numbers) and also unequal to each other.

When the discriminant of the quadratic equation equal to 0, then the roots will become equal to each other and would also be rational in nature.

If one root of a quadratic equation is a surd than the other root of that quadratic equation would be the conjugate of that particular surd.

# Making New Quadratic Equations when the roots of the original equation are changed -

If the roots are inverted that is their reciprocal becomes the root of the equation then in the original quadratic equation we replace the variable x by 1/x.

If a particular constant is added to both the roots of the quadratic equation then in the new quadratic equation the variable gets reduced by the same constant.

If a particular constant is reduced to both the roots of the quadratic equation then in the new quadratic equation the variable gets increased by the same constant.

# Sum and product of roots taken one at a time, two at a time, three at a time and so on has been discussed for polynomials having degree more than two. After that I have taken up the graphical representation of quadratic equation for different ranges of the discriminant and for positive and negative values of the constant a.

# Lastly I have taken up the concept of Maximum and Minimum value of a quadratic equation. First I have explained that how we would find out whether the maximum or minimum value exists and then the formula for calculating that value has been proved by me.

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