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Quadratic Equations - Coggle Diagram
Quadratic Equations
Discriminant: D = b2 - 4ac
D > 0, the roots are real and distinct
D = 0, the roots are real and equal.
D < 0, the roots do not exist or the roots are imaginary.
What is Quadratic Equation?
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).
Quadratic Equation Formula
The solution or roots of a quadratic equation are given by the quadratic formula:
(α, β) = [-b ± √(b2 – 4ac)]/2a
Important Formulas for Solving Quadratic Equations
The following list of important formulas is helpful to solve quadratic equations.
The quadratic equation in its standard form is ax2 + bx + c = 0
The discriminant of the quadratic equation is D = b2 - 4ac
For D > 0 the roots are real and distinct.
For D = 0 the roots are real and equal.
For D < 0 the roots do not exist, or the roots are imaginary.
The formula to find the roots of the quadratic equation is x = [-b ± √(b² - 4ac)]/2a.
The sum of the roots of a quadratic equation is α + β = -b/a = - Coefficient of x/ Coefficient of x2.
The product of the Root of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x2
The quadratic equation having roots α, β, is x2 - (α + β)x + αβ = 0.
The condition for the quadratic equations a1x2 + b1x + c1 = 0, and a2x2 + b2x + c2 = 0 having the same roots is (a1b2 - a2b1) (b1c2 - b2c1) = (a2c1 - a1c2)2.
For positive values of a (a > 0), the quadratic expression f(x) = ax2 + bx + c has a minimum value at x = -b/2a.
For negative value of a (a < 0), the quadratic expression f(x) = ax2 + bx + c has a maximum value at x = -b/2a.
For a > 0, the range of the quadratic equation ax2 + bx + c = 0 is [b2 - 4ac/4a, ∞)
For a < 0, the range of the quadratic equation ax2 + bx + c = 0 is : (∞, -(b2 - 4ac)/4a]
Roots of Quadratic Equation
The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0.
The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.
One of the roots of the quadratic equation is zero and the other is -b/a if c = 0
Both the roots are zero if b = c = 0
The roots are reciprocal to each other if a = c
Nature of Roots of the Quadratic Equation
The nature of roots of a quadratic equation can be found without actually finding the roots (α, β) of the equation. This is possible by taking the discriminant value, which is part of the formula to solve the quadratic equation. The value b2 - 4ac is called the discriminant of a quadratic equation and is designated as 'D'. Based on the discriminant value the nature of the roots of the quadratic equation can be predicted.