Please enable JavaScript.
Coggle requires JavaScript to display documents.
Quadratic Equations. - Coggle Diagram
Quadratic Equations.
-
The sum of the roots of a quadratic equation is α + β = -b/a = - Coefficient of x/ 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]
-
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.
Quadratic Formula Proof
Consider an arbitrary quadratic equation: ax2 + bx + c = 0, a ≠ 0
To determine the roots of this equation, we proceed as follows:
-
Now, we express the left hand side as a perfect square, by introducing a new term (b/2a)2 on both sides:
-
What is Quadratic Equation? A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.
-
-
-
-
For D < 0 the roots do not exist, or the roots are imaginary.
Roots of a Quadratic Equation
The roots of a quadratic equation are the two values of x, which are obtained by solving the quadratic equation. The roots of a quadratic equation are referred to by the symbols alpha (α), and beta (β).
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.
Relationship Between Coefficients and Roots of Quadratic Equation
The coefficient of x2, x term, and the constant term of the quadratic equation ax2 + bx + c = 0 are useful to study more about the properties of roots of the quadratic equation.
Sum of the Roots: α + β = -b/a = - Coefficient of x/ Coefficient of x2
Product of the Roots: αβ = c/a = Constant term/ Coefficient of x2
Methods to Solve Quadratic Equations
A quadratic equation can be solved to obtain two values of x or the two roots of the equation
Factorization of Quadratic Equation
Factorization of quadratic equation follows a sequence of steps. For a general form of the quadratic equation ax2 + bx + c = 0, we need to first split the middle term into two terms, such that the product of the terms is equal to the constant term.
Quadratic Formula to Find Roots
The quadratic equations which cannot be solved through the method of factorization can be solved with the help of a formula.
Method of Completing the Square
The method of completing the square for a quadratic equation, is to algebraically square and simplify, to obtain the required roots of the equation. Consider a quadratic equation ax2 + bx + c = 0, a ≠ 0.
Graphing a Quadratic Equation
The graph of the quadratic equation ax2 + bx + c = 0 can be obtained by representing the quadratic equation as a function y = ax2 + bx + c.
Quadratic Equations Having Common Roots
Let the two quadratic equations having common roots are a1x2 + b1x + c1 = 0, and a2x2 + b2x + c2 = 0. Let us solve these two equations to find the conditions for which these equations have a common root.
Solving a Quadratic Equation - Tips and Tricks
Some of the below-given tips and tricks on quadratic equations are helpful to more easily solve quadratic equations.