Quadratic Equations - Sibanand Pattnaik


  • QA/DILR Mentor | Be Legend


    Definition

    Any equation of degree 2 is known as a quadratic equation.
    General form is ax^2+bx + c = 0
    The numbers a, b are called the coefficients of this equation and c is the constant.

    Roots

    The possible values of x which satisfy the quadratic equation are called the roots of the quadratic equation. Quadratic equation will have two roots either real or imaginary. We normally denote them as α and β.
    A root of the quadratic equation is a number such that pα ^2 + qα + r = 0 or pβ^ 2 + qβ + r = 0.

    If α & β are roots of the quadratic equation ax^2+bx + c = 0, then (x - α) and (x -β) are factors of ax^2+bx + c = 0
    OR ax^2+bx + c = (x - α)(x -β)

    Properties of roots

    f(x) = ax^2 + bx + c = 0
    The sum of the roots= α + β = -b/a.
    The product of roots = α β = c/a
    If α and β are the roots of the equation ax^2 + bx + c = 0, then we can write
    ax^2 + bx + c = x^2 - (α + β) x + αβ = x^2 + (sum of the roots) x + product of the roots = 0
    Or, ax^2 + bx + c = a (x – α) (x – β) = 0

    Discriminant

    Given is the quadratic equation ax^2 + bx + c = 0, where a ≠ 0.
    (b^2 – 4ac) is also known as Discriminant (D)
    If D = 0, then √(b^2-4ac)= 0. So, the roots will be real and equal.
    If D > 0, then √(b^2-4ac)> 0. So, the roots will be real and distinct.
    If D < 0, then √(b^2-4ac) is not real. So, the roots will not be real.
    If D is a perfect square (including D = 0) and a, b and c are rational, then the roots will also be rational.

    Descartes' Rule of Signs of Roots

    The maximum number of positive roots of any equation is equal to the change of signs from positive (+ve) to negative (-ve) and from negative (-ve) to positive (+ve).

    Solving by factorization

    Ax^2 + Bx + C =0

    We have to write B as the sum of 2 numbers say P and Q such that the product of P and Q is equal to the product of A and C
    B = (P + Q)
    A * C = P * Q

    For example
    5x^2 -2x -4 = 0
    So P * Q = 5 * (-4) = -20
    And P + Q = -2
    By trial and error we get - 10 and 2
    5x^2 -10x + 2x -4 = 0
    5x(x-2) + 2(x-2) =0
    OR (5x+2)(x-2)=0

    Solving by Formula

    Assuming that α and β are the roots of the equation ax^2 + bx + c = 0, where a ≠ 0.
    Then α = (-b+√(b^2-4ac))/2a and β = (-b-√(b^2-4ac))/2a

    Minimum Value of Quadratic Equations

    Ax^2 + Bx + C = 0
    OR (sqrtA * x)^2 + (2 * (sqrtA * x) * B/(2 * (sqrtA)) ) + (B/(2 * (sqrtA * x)^2 + C = 0
    OR (sqrtA * x + (B/(2 * (sqrtA)) ^2 + (C - (B/(2 * (sqrtA * x)^2 ) = 0
    Now (sqrtA * x + (B/(2 * (sqrtA*x)) ^2 will always be positive as it’s a square term so its minimum value will be 0
    SO the minimum value of the expression (sqrtA * x + (B/(2 * (sqrtA * x)) ^2 + (C - (B/(2 * (sqrtA * x)^2 ) is equal to:
    (C - (B/(2 * (sqrtA * x)^2 )
    And this value is achieved when (sqrtA * x + (B/(2 * (sqrtA)) is 0
    Or x = -B/2A
    So the minimum value of a quadratic equation is (C - (B/(2 * (sqrtA * x)^2 ))
    And this value is achieved when x = -B/2A

    Graph Basics

    When A (coefficient of x^2) is positive
    We cant find the minimum value but not the maximum value
    The above post also helps us relate quadratic equation to graphs:
    We all know the graph of a quadratic equation is a parabola
    Graph of (x+a)^2 will be same, only shift a place to the left on x axis
    Graph of (x-a)^2 will be same, only shift a place to the right on x axis
    Graph of (x)^2 + b will be same, only shift b place up on y axis
    Graph of (x)^2 - b will be same, only shift b place down on y axis
    We can therefore draw the graph of the quadratic equation say (x^2 - 4x - 12) = (x-2)^2 – 16
    So the graph will be graph of x^2 shifted 2 places to the right on x axis and 16 places down on the y axis
    Its lowest point will be when y = -16 and x = 2
    Also in the graph of (x^2 - 4x - 12), when x = 0, y = -12 so the graph will cut the y axis when y =-12
    Also (x^2 - 4x - 12) = (x-6)(x+2)
    So the graph will cut x axis at 6 and -2
    So one positive and one negative root which is also given by the sign changes
    So the graph will go down from 2nd quadrant, cut x axis at -2 and enter the 3rd quadrant , then cut y axis at -12 and enter the 4th quadrant, then reach its lowest point when y =-16 and x=2 and then go up and cut x axis at 6 and into the 1st quadrant
    So if the value of c is positive therefore the lowest point of the graph is positive so the graph never reaches x axis so there are no real roots (all this is when the graph is U shaped i.e. a is positive)

    When A (coefficient of x^2) is negative
    We cant find the maximum value but not the minimum value
    Everything will be same as above except in the inverted form
    So if C is negative so the equation will not have any real roots

    Also check the below videos for some good concepts


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