*This article is about algebraic equations of degree two and their solutions. For functions defined by polynomials of degree two, see *Quadratic function*.* In algebra, a **quadratic equation** (from the Latin*quadratus* for “square”) is any equation that can be rearranged in standard form as ax^{2} + bx + c.

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- In order to solve a quadratic equation of the form ax
^{2}+ bx + c, we first need to calculate the discriminant with the help of the formula D = b^{2 }– 4ac. - The solution of the quadratic equation ax
^{2}+ bx + c= 0 is given by x = [-b ± √ b^{2 }– 4ac] / 2a - If α and β are the roots of the quadratic equation ax
^{2}+ 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 ax^{2} + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:

- D > 0 iff the roots are real and distinct i.e. the roots are unequal
- D = 0 iff the roots are real and coincident i.e. equal
- D < 0 iffthe roots are imaginary
- 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 ax^{2} + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then

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

x^{2} – (Sum of roots)x + (Product of roots) = 0. So if α and β are the roots of equation then the quadratic equation is x^{2} – (α + β)x + α β = 0

- For the quadratic expressiony = ax
^{2}+ bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.

- If a > 0, then the shape of the parabola is concave upwards
- 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 = ax
^{2}+ bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.

- y ∈[(4ac-b
^{2}) / 4a, ∞] if a > 0 - If a < 0, then y ∈ [-∞, (4ac-b
^{2}) / 4a]

- The quadratic function of the form f(x, y) = ax
^{2}+by^{2}+ 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af^{2}– bg^{2}– ch^{2}= 0 - In general, if α
_{1},α_{2}, α_{3}, …… ,α_{n}are the roots of the equation

f(x) = a_{0}x^{n} +a_{1}x^{n-1} + a_{2}x^{n-2} + ……. + a_{n-1}x + a_{n}, then 1.Σα_{1 }= – a_{1}/a_{0} 2.Σ α_{1}α_{2} = a_{2}/a_{0} 3.Σ α_{1}α_{2}α_{3} = – a_{3}/a_{0} ……… ………. Σ α_{1}α_{2}α_{3} ……α_{n}= (-1)^{n} a_{n}/a_{0}

- 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.