Revision Notes on Quadratic Equations

• In order to solve a quadratic equation of the form ax² + bx + c, we first need to calculate the discriminant with the help of the formula D = b² – 4ac.

• The solution of the quadratic equation ax² + bx + c= 0 is given by x = [-b ± √ b² – 4ac] / 2a

• If α and β are the roots of the quadratic equation ax² + bx + c = 0, then we have the following results for the sum and product of roots:

α + β = -b/a

α.β = c/a

α – β = √D/a

• It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.

• Nature of Roots:

Consider an equation ax² + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:

1. D > 0 iff the roots are real and distinct i.e. the roots are unequal

2. D = 0 iff the roots are real and coincident i.e. equal

3. D < 0 iffthe roots are imaginary

4. The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.

Consider an equation ax² + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then

1. If D > 0 and is also a perfect square then the roots are rational and unequal.

2. If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.

• If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula

x² – (Sum of roots)x + (Product of roots) = 0.

So if α and β are the roots of equation then the quadratic equation is

x² – (α + β)x + α β = 0

• For the quadratic expressiony = ax² + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.

1. If a > 0, then the shape of the parabola is concave upwards

2. If a < 0, then the shape of the parabola is concave upwards

• Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.

• The maximum and minimum values of the expression y = ax² + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.

1.  y ∈[(4ac-b²) / 4a, ∞] if a > 0

2.  If a < 0, then y ∈ [-∞, (4ac-b²) / 4a]

• The quadratic function of the form f(x, y) = ax²+by² + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af² – bg² – ch² = 0

• In general, if α₁,α₂, α₃, …… ,α_n are the roots of the equation

f(x) = a₀xⁿ +a₁x^n-1 + a₂x^n-2 + ……. + a_n-1x + a_n, then

1.Σα₁ = - a₁/a₀

2.Σ α₁α₂ = a₂/a₀

3.Σ α₁α₂α₃ = - a₃/a₀

………   ……….

  Σ α₁α₂α₃ ……α_n= (-1)ⁿ a_n/a₀

• Every equation of n^th degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.

• If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.

• Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.