Quadratic Equations

An equation whose highest power is 2 is an quadratic equation. The graph of an quadratic is always a parabola.

Solution of a Quadratic Equation
A Quadratic Equation cannot have more than two roots.

Formula Method
x =[-b ±√(b.2.-4ac)] / 2a

Substitution Method
Form aZ +b/Z + c

Form (x + a)(x + b)(x + c)(x + d) + k = 0

Step 1 :De-factorize (2 at a time)

Step 2 : Substitute  [ if a + b = c+ d ]

Step 3 : Solve and Re-substitute

Complex & Irrational Quadratic Equations
If any coefficient is a non-real complex and p + iq is a complex root of equation, then p - iq need not b the other root.

If any coefficient is an irrational number and p + √q is a complex root of equation, then p - √q need not b the other root.

The above holds true for even higher degree polynomials.

Relation between Roots and Coefficients
α + β = - b/a

α β = c/a

Nature of Roots
b2 - 4ac > 0              Roots are Real and Unequal

b2 - 4ac = 0              Roots are Real and '''Equal. The equation is a perfect square '''

b2 - 4ac < 0              Roots are Complex

b2 - 4ac is not a perfect square

Roots are Real and Irrational 

Sum and Product of Roots
α + β = - b/a

αβ = c/a

Formation of Quadratic Equation
x2- (Sum of Roots)x + (Product of Roots) = 0

Sign of Roots
{Based on laws of inquations}

Positive,  if both sum and product of roots are positive.

Negative,  if both sum and product of roots are negative.

Opposite Signs,  if product of roots is negative.

Equal Magnitude and Opposite Signs,  if Sum of Roots is 0.

Graph
Parabolic Graph

b2 = 4ca   y2 = 4ax

Quadratic Inequalities
Quadratic Inequalities are solved in the same way as quadratic equations (Interpretation is important).

Quadratic & Higher Order Polynomials
A polynomial of n order, has n roots. Note the similarities in quadratic equations and higher order polynomials.

Consider α β γ to be roots of third order polynomial ax3 + bx2+ cx + d ; then

α+β+γ = -b/a

αβ + βγ + αγ = c/a

αβγ = -d / a

Consider α β γ δ to be the roots of a fourth order polynomial ax4 + bx3 + cx2 + dx + e ; then

α+β+γ+δ = -b/a

αβ + βγ + γδ + αγ + αδ + βδ = c/a

αβγ+ βγδ + αγδ + αβδ = -d/a

αβγδ = e/a

Common Roots in two Quadratic Equations
α is a common root in a1x2+b1x+c1 = 0 and a1x2+b1x+c1 = 0; thus α satisfies both equations.

If α β are the roots of first equation ; α γ are the roots of second equation ; then

Sum and Products of roots of both equations can be equated due to the common root α. (This helps in solving such equations)