state ways to solve liner ,quadratic , and cubic polynomial
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Step-by-step explanation:
Quadratic Equation in One Variable
An equation in which the variable varies to a degree of two is a quadratic equation. In other words, an equation in which the variable has the maximum degree of two is a quadratic one. Quadratic means related to a square.
The Standard form of Quadratic Equation
The general form of a quadratic equation is a x2 + b x + c = 0. Here, x is a variable and a, b, and c are constants with a ≠ 0. The number of solutions that satisfy a quadratic equation will be two. A graph of a quadratic equation is a parabola.
cubic
Methods for Solving Quadratic Equations
I. Factorization Method
The main principle of this method is the principle of zero products. According to it if the product of two numbers is zero, then at least one of the factors is zero.
If α and β are the two roots of the quadratic equation, a x2 + b x + c = 0. These roots are the factors of the equation in a way such that the sum of the roots is equal to the negative of the constant ba in the equation. The constant ca is equal to the product of the roots.
In this method, we need to find the factors which on being added and multiplied will result in the respective constants of the equations. The quadratic equation is thus written as, x2 + (sum of the root) x + (product of the roots) = 0.
II. Method of Perfect Square
In this method, we try to reduce the quadratic equation into a perfect square. The steps for solving the equations are
If the quadratic equation is of the form a x2 + b x + c = 0. Divide both sides of the equations by a
Try to make the left-hand side a complete or a perfect square by adding the value of (b2a)2 in the left. Add the right-hand side by the same (b2a)2
Take the square root of both sides
Solve the like terms in the equation to find the value(s) of x. This will give the roots of the variable
III. Using Quadratic Formula
This method is the direct application of the method of the perfect square. The roots of the equation are obtained directly using the formula
cubic
Putting the values in the above formula we find two roots, one for the positive value of the square root and one for the negative value of the square root.
Nature of the Roots
Let us denote √(b2 − 4ac) by D. Here, D is the discriminant of the equation. D determines the nature of the roots.
If D = 0, the roots are real and equal
If D < 0, then the roots are imaginary
The roots are real and distinct if D > 0
For D > 0, if D is a perfect square also then the roots are real, rational and unequal
If D > 0 but not a perfect square, then the roots are real, irrational and distinct