Math, asked by Anonymous, 4 months ago

What is Quadratic equation

Answers

Answered by pazhaniakshaiadhi

Step-by-step explanation:

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c where a, b, c, ∈ R and a ≠ 0. 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).

General from: ax2 + bx + c = 0

Examples: 3x2 + x + 5 = 0, -x2 + 7x + 5 = 0, x2 + x = 0.

1. The roots of the quadratic equation: x = (-b ± √D)/2a, where D = b2 – 4ac

2. Nature of roots:

D > 0, roots are real and distinct (unequal)
D = 0, roots are real and equal (coincident)
D < 0, roots are imaginary and unequal

3. The roots (α + iβ), (α – iβ) are the conjugate pair of each other.

4. Sum and Product of roots: If α and β are the roots of a quadratic equation, then

S = α+β= -b/a = coefficient of x/coefficient of x2
P = αβ = c/a = constant term/coefficient of x2
5. Quadratic equation in the form of roots: x2 – (α+β)x + (αβ) = 0

6. The quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have;

One common root if (b1c2 – b2c1)/(c1a2 – c2a1) = (c1a2 – c2a1)/(a1b2 – a2b1)
Both roots common if a1/a2 = b1/b2 = c1/c2
7. In quadratic equation ax2 + bx + c = 0 or [(x + b/2a)2 – D/4a2]

If a > 0, minimum value = 4ac – b2/4a at x = -b/2a.
If a < 0, maximum value 4ac – b2/4a at x= -b/2a.

8. If α, β, γ are roots of cubic equation ax3 + bx2 + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a

9. A quadratic equation becomes an identity (a, b, c = 0) if the equation is satisfied by more than two numbers i.e. having more than two roots or solutions either real or complex

Answered by anjali5087

Answer: