Quadratic Equations, Cubic and Higher Order Equations
Introduction
Quadratic equations are those equations which can be written in the form f(x)=0 where f(x) is a second degree polynomial. General form of a Quadratic equation is: ax2
x = (-b + √D ) / 2a
and
x = (-b - √D ) / 2a
Where D is the Discriminant and D = b2 - 4ac
The Discriminant
The term b2- 4ac
How do we plot the curve on a graph paper?
For various values of 'x', evaluate the expression f(x) similar to the way in which algebraic expressions in one variable are evaluated.
There are plenty of examples for evaluating algebraic polynomials for specific values of 'x': here and here. In the case of quadratic polynomials, we get a parabolic curve.
Roots of a Quadratic Equation : Are they real/unreal, equal or unequal ?
If D>0, the equation has real and unequal roots, if D=0, the equation has real and equal roots (also called real repeated roots), and if D<0, the equation has unreal roots, occurring as conjugate pairs. That is if one root is of
the form u+iv, the other root would be u-iv. If α and β are roots of a Quadratic equation, then
The equation can be written as : a(x-α)(x-β)=0
=> The equation could also be written as x-(α+β)+αβ=0
Using Substitution to convert equations to Quadratic Form
A Quadratic equation could be solved by factorization, or by using the direct formula written above. Certain equations are not quadratic, but can be reduced to a quadratic form by certaing substitutions. In such cases, applying the right form of substitution yields the required solutions.
Example
3x+√x-2=0;
x+√(x-4)=4;
Remember, when using substitutions, be sure that you solve for the original variable, and that the solution does not violate any constraints. For instance, if you have √(x-1)
Also, if you have a step like: (x-z)(f(x))=(x-z)(g(x))
Dealing with Cubic and Higher Order Equations
A cubic equation is of the form f(x)=0, where f(x) is a degree 3 polynomial. The general form of a cubic equation is ax3+bx2+cx+d=0, where a is not equal to 0. If α, β, γ are roots of the equation, then equation could be written as:
a(x-α)(x-β)(x-γ)=0, or also as
x3-(α+β+γ)x2+(αβ+βγ+γα)x-(αβγ)=0
Thus, we have
α+β+γ=-b/a
αβ+βγ+γα=c/a
αβγ=-d/a
A quadratic equation may have all repeated real roots, two repeated and one distinct real root,
one distinct real and two conjugate unreal roots, all distinct real roots.
Higher Order equations - Relationship between roots
Equations of the type f(x)=0 where degree of f(x) is greater than 3 are generally termed as Higher Order equations
The number of roots of an equation with real coefficients is equal to the degree of f(x). The way to solve a higher order equation is by factorization, or by using the factor theorem, or by reducing it to one of the lower order equations.
The factor theorem is: (x-a) is a factor of f(x) if f(a)=0.
The relationship between the coefficients and the roots can be explained by an example. Consider a fifth degree equation: ax5432
Sum of roots = -b/a
Sum of pairwise products of roots=c/a
Sum of products of roots taken three at a time = -d/a
Sum of products of roots taken four at a time = e/a
Product of roots=-f/a
In general, for a n degree polynomial equation in variable x, sum of products of roots taken m at a time is (-1)m.Coefficient of xn-m/Coefficient of xn
Here are some of the examples and problems solved in the tutorial :
- Solving basic quadratic equations
- Using factorization or formula method for finding the roots
- Will roots be equal unequal or complex? Identifying this using the discriminant (D < 0, D = 0, D > 0 and so on)
- being able to plot the parabolic curve representing the quadratic equation and identifying roots from the plots or graph
- Write an equation whose roots are provided
- If α,β are roots of a quadratic equation find their sum and product.
- Find the number of real roots of an equation.
- Find the number of solutions of a given equation.
- Questions related to simple cubic equations.
