Quadratic Equations

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 → Complex (imaginary) roots
• D = 0 → Real, equal roots: x = −b/2a
• D > 0, perfect square → Real, rational, unequal
• D > 0, not a perfect square → Real, irrational, unequal (conjugate surds)

If roots are irrational or complex, they always occur in conjugate pairs

The General Quadratic equation will be in the form of a$$x^{2}$$+b$$x$$+c = 0

The values of ‘x’ satisfying the equation are called the roots of the equation.

The value of roots, p and q = $$\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$

The sum of the roots = p+q = $$\dfrac{-b}{a}$$

Product of roots = p*q = $$\dfrac{c}{a}$$

If c and a are equal then the roots are reciprocal to each other.

If b = 0, then the roots are equal and are opposite in sign.

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 $$A_{n}X^{n}$$ + $$A_{n-1}X^{n-1}$$ + ... + $$A_{1}X$$ + $$A_{0}$$ = 0 and n>=3, 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 =$$(-1)^nA_0/A_n$$

Ex: For a cubic ax³+bx²+cx+d=0 with roots p,q,r:

• p+q+r = −b/a

• pq+qr+rp = c/a

• pqr = −d/a

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

• If a > 0: minimum value = $$\frac{4ac - b^2}{4a}$$ and occurs at x = $$\frac{-b}{2a}$$

• If a < 0: maximum value = $$\frac{4ac - b^2}{4a}$$ and occurs at x = $$\frac{-b}{2a}$$

Signs of the roots: Let P be the 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

Finding a quadratic equation:

• 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 $$a(y/k)^2 + b(y/k) + c = 0$$

• If roots are negatives of roots of ax²+bx+c=0, new equation is ax²−bx+c=0

• If p and q are roots of ax²+bx+c=0, and we want a new equation whose roots are p² and q². The equation will be $$a^2x^2−(b^2−2ac)x+c^2=0$$