![]() ![]() Hence the roots will be real and distinct.įind the value of m for which the equation (1 m)x 2 –2(1 3m)x (1 8m) = 0 has equal roots. A quadratic equation will always have two roots. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x). Prove that the roots of the quadratic equation ax 2 – 3bx – 4a = 0 are real and distinct for all real and a and b.ĭ = (–3b) 2 –4(–4a) (a) = 9b 2 16a 2 which is always positive. Quadratic equations are the polynomial equations of degree 2 in one variable of type: f (x) ax 2 bx c where a, b, c, R and a 0. Then a b = –b/a and ab = c/a.Ī quadratic equation, whose roots are a and b can be written as (x – a) Q Let a and b be two roots of the given quadratic equation. Q If the quadratic equation is satisfied by more than two numbers (real or complex), then it becomes an identity i.e. ![]() If a = 1 and b, c are integers and the roots of the quadratic equation are rational, then the roots must be integers. Q The quadratic equation has rational roots if D is a perfect square and a, b, c are rational. Q If p is an irrational root of the quadratic equation, then p – is also root of the quadratic equation provided that all the coefficients are rational. Q If p iq (p and q being real) is a root of the quadratic equation where i =, then p – iq is also a root of the quadratic equation. Q The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 i.e. Q The quadratic equation has real and distinct roots if and only if D > 0 i.e. Q The quadratic equation has real and equal roots if and only if D = 0 i.e. Q The quantity D(D=b 2 – 4ac) is known as the discriminant of the quadratic equation. So (a – ib) is the other solution of the equation which is complex conjugate of the first root (a ib). (a – ib) is also a root of the equation ax 2 bx c = 0 and we know that every quadratic equation can have two and only two solution. Or (aa 2 – 2aabi – ab 2) ba bbi c = 0 Or a(a 2 2abi – b 2) ba ibb c = 0 Let a ib is one root of the quadratic equation ax 2 bx c = 0 ![]() If a quadratic equation has one complex root then the other must be complex conjugate of the first root. any type of quadratic equation can be solved.Ĭomplex roots of a quadratic equation always occur in pair In this way, we can know the nature of the roots of the given equation i.e. Here both the roots of the equation would be rational in nature. b 2 – 4ac > 0 Þ would be a real number say (K). These are the roots of the equation ax 2 bx c = 0. We have f(x) = ax 2 bx c where a, b, c Î R. If a, b, c Î Q (set of rational roots are real, different and rational number) and D is perfect irrational square If a, b, c Î Q (set of rational roots are real, different and rational number) and D is perfect square If you know the value of D of a given quadratic equation, you can be certain about the nature of its roots. b 2 – 4ac) discriminates the nature of root and so it is called discriminant (D) of the quadratic equation i.e.ĭ = b 2 – 4ac. The most effective way to solve a quadratic equation is to use the quadratic formula.
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