The theory of quadratic equation formulae will help us to solve different types of problems on quadratic equation.

The general form of a quadratic equation is ax\(^{2}\) + bx + c = 0 where a, b, c are real numbers (constants) and a ≠ 0, while b and c may be zero.

**(i)** The Discriminant of a quadratic equation is ax\(^{2}\) + bx + c = 0 (a ≠ 0) is ∆ = b\(^{2}\) - 4ac

**(ii)** If α and β be the roots of the equation ax\(^{2}\) + bx + c = 0 (a ≠ 0) then

α + β = -\(\frac{b}{a}\) = -\(\frac{coefficient of x}{coefficient of x^{2}}\)

and αβ = \(\frac{c}{a}\) = \(\frac{constant term}{coefficient of x^{2}}\)

**(iii)** The formula for the formation of the quadratic equation
whose roots are given: x^2 - (sum of the roots)x + product of the roots = 0.

**(iv)** When a, b and c
are real numbers, a ≠ 0 and discriminant is positive
(i.e., b\(^{2}\) - 4ac > 0), then the roots α and β of
the quadratic equation
ax\(^{2}\) + bx + c = 0 are
real and unequal.

** ****(v)** When a, b and c are real
numbers, a ≠ 0 and discriminant is zero (i.e., b\(^{2}\) -
4ac = 0), then the roots α and β of the quadratic
equation ax\(^{2}\) + bx + c = 0 are
real and equal.

** ****(vi)** When a, b and c are real
numbers, a ≠ 0 and discriminant is negative (i.e., b\(^{2}\) -
4ac < 0), then the roots α and β of the quadratic
equation ax\(^{2}\) + bx + c = 0 are
unequal and imaginary. Here the roots α and β are a pair of the complex
conjugates.

** ****(viii)** When a, b and c are real
numbers, a ≠ 0 and discriminant is positive and perfect square,
then the roots α and β of the quadratic
equation ax\(^{2}\) + bx + c = 0 are
real, rational unequal.

** ****(ix)** When a, b and c are real
numbers, a ≠ 0 and discriminant is positive but not a perfect
square then the roots of the quadratic
equation ax\(^{2}\) + bx + c = 0 are
real, irrational and unequal.

** ****(x)** When a, b and c are real
numbers, a ≠ 0 and the discriminant is a perfect square but any
one of a or b is irrational then the roots of the quadratic equation
ax\(^{2}\) + bx + c = 0 are
irrational.

**(xi)** Let the two quadratic equations
are a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0

**Condition for one common root:** (c1a2 - c2a1)^2 = (b1c2 - b2c1)(a1b2 - a2b1), which is the
required condition for one root to be common of two quadratic equations.

**Condition for both roots common: **a1/a2 = b1/b2 = c1/c2

**(xii)** In a quadratic equation with
real coefficients has a complex root α + iβ then it has also the conjugate
complex root α - iβ.

**(xiii)** In a quadratic equation with
rational coefficients has a irrational or surd root α + √β, where α and β
are rational and β is not a perfect square, then it has also a conjugate root α
- √β.

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