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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}â€“ 4ac. - The solution of the quadratic equation ax
^{2}+ bx + c= 0 is given by x = [-b Â± âˆš b^{2}â€“ 4ac] / 2a - If Î± and Î² 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:

Î± + Î² = -b/a

Î±.Î² = c/a

Î± â€“ Î² = âˆšD/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 âˆˆ R and a â‰ 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 âˆˆ R and i = âˆš-1.

Consider an equation ax^{2}+ bx + c = 0, where a, b and c âˆˆQ and a â‰ 0, then

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

2. If Î± = p + âˆšq is a root of the equation, where â€˜pâ€™ is rational and âˆšq is a surd, then the other root must be the conjugate of it i.e. Î² = p - âˆšq 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}â€“ (Sum of roots)x + (Product of roots) = 0.

So if Î± and Î² are the roots of equation then the quadratic equation is

x^{2}â€“ (Î± + Î²)x + Î± Î² = 0 - For the quadratic expression y = ax
^{2}+ bx + c, where a, b, c âˆˆ R and a â‰ 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 = ax
^{2 }+ bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.

1. y âˆˆ[(4ac-b^{2}) / 4a, âˆž] if a > 0

2. If a < 0, then y âˆˆ [-âˆž, (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 â€“af^{2 }â€“ bg^{2}_{ }â€“ ch^{2}= 0 - In general, if Î±
_{1},Î±_{2}, Î±_{3}, â€¦â€¦ ,Î±_{n}are the roots of the equation

f(x) = a_{0}x^{n}+a_{1}x^{n-1}+ a_{2}x^{n-2}_{ }+ â€¦â€¦. + a_{n-1}x + a_{n}, then

1.Î£Î±_{1}= - a_{1}/a_{0}

2.Î£ Î±_{1}Î±_{2}= a_{2}/a_{0}

3.Î£ Î±_{1}Î±_{2}Î±_{3}= - a_{3}/a_{0}

â€¦â€¦â€¦ â€¦â€¦â€¦.

Î£ Î±_{1}Î±_{2}Î±_{3 }â€¦â€¦Î±_{n}= (-1)^{n}a_{n}/a_{0} - Every equation of n
^{th}degree has exactly n roots (n â‰¥1) and if it has more than n roots then the equation becomes an identity. - If there are two real numbers â€˜aâ€™ and â€˜bâ€™ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between â€˜aâ€™ and â€˜bâ€™.
- Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.

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