Quadratic Equations

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Theory

Let D denote the discriminant $$b^{2}-4ac$$. Hence, depending on the sign and value of D, nature of the roots would be as follows:

  • D<0 and abs(D) is not a perfect square: Roots are complex and irrational. They can be represented as p+iq and p-iq where p and q are the real and imaginary parts of the complex roots. p is rational and q is irrational.

  • D < 0 and abs(D) is a perfect square: Roots are complex but rational. They can be represented as p+iq and p-iq where p and q are both rational.

  • D=0 : Roots are real and equal. X = -b/2a

  • D>0 and D is not a perfect square: Roots are conjugate surds

  • D>0 and D is a perfect square: Roots are real, rational and unequal

Formula

Sum of the roots = -b/a

Product of roots = c/a

Formula

If $$A_{n}X^{n}$$ + $$A_{n-1}X^{n-1}$$ + ... + $$A_{1}X$$ + $$A_{0}$$ = 0, then

  • Sum of the roots = $$-A_{n-1}/ A_{n}$$

  • Sum of roots taken two at a time = $$A_{n-2}/ A_{n}$$

  • Sum of roots taken three at a time = $$-A_{n-3}/ A_{n}$$ and so on

  • Product of the roots = $$(-1)^{n}-A_{0}/ A_{n}$$

Theory

Minimum and maximum values of $$ax^{2}+bx+c=0$$ :

  • If a > 0: minimum value = $$(4ac - b^2)/4a$$ and occurs at x = -b/2a

  • If a < 0: maximum value = $$(4ac - b^2)/4a$$ and occurs at x = -b/2a

Theory

Signs of the roots: Let P be product of roots and S be their sum

  • P>0, S>0 : Both roots are positive

  • P>0, S<0 : Both roots are negative

  • P<0, S>0 : Numerical smaller root is negative and the other root is positive

  • P<0, S<0 : Numerical larger root is negative and the other root is positive

Shortcuts

Finding a quadratic equation:

  • If roots are given : (x-a)(x-b)=0 => $$x^2 - (a+b)x + ab = 0$$

  • If sum s and product p of roots are given: $$x^2 - sx + p = 0$$

  • If roots are reciprocals of roots of equation $$ax^2 + bx + c = 0$$, then equation is $$cx^2 + bx + a = 0$$

  • If roots are k more than roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y-k)^2 + b(y-k) + c = 0$$

  • If roots are k times roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y/k)^2 + b(y/k) + c = 0$$

Theory

Descartes Rules : A polynomial equation with n sign changes can have a maximum of n positive roots. To find the maximum possible number of negative roots, find the number of positive roots of f(-x).

An equation where highest power is odd must have at least one real root

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