How to Solve Quadratic Equations with Common Roots?

Solving the quadratic equation with one common root depends on the calculation of the same coefficient of x2 of the above equations and subtracting the two equations. Moreover, the equations for the other roots can be determined by identifying the relations between the coefficient and roots of the equations. Quadratic equation with a common root is a polynomial and can be determined by writing in the factor (x-a) (x-b) (x-c)…… for any value of a,b,c. Therefore, the equation with a common root will be determined by the multiplicity of 2.

What is a Quadratic Equation?

Quadratic equation refers to the second degree algebraic equation and the general terms of the equation are defined below:

ax2 + bx + c = 0 (a,b are termed as the coefficient, x is denoted as variable and C is denoted as the constant)

From the above equation, it is clear that the quadratic equation is the equation of degree 2. The primary property or condition of the quadratic equation is that the coefficient of x2 is a non-zero term. 

Determine Whether the Following Equation is Quadratic Equation

Lert α, β is the common root of the below equation:

a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0, therefore from the equation it can be derived that

α + β = -b1/a1, α β = c1/a1 and α + β = -b2/a2, α β = c2/a2

Hence from the above equation it can be derived that -b/a1 = – b2/a2 and c1/a1 = c2/a2

⇒ a1/a2 = b1/b2 and a1/a2 = c1/c2

⇒ a1/a2 = b1/b2 = c1/c2

The above equation is the general condition for determining the quadratic equation with common roots.

What Is The Standard Form of A Quadratic Equation?

The standard form of the quadratic equation is defined as ax2 + bx + c = 0 (Where a and b is coefficient, x is the variable and c is the constant). The function of the quadratic equation is  f(x) = a(x – h)² + k and it is not equal to the zero. If the value of the a is positive then the position of the graphs will be in the upward direction. Moreover, if the value of a is negative, then the position of the graph will be in a downward direction.

Explain the Properties of Quadratic Equation

ax2 + bx + c = 0

The above standard form of the quadratic equation with complex coefficient and rational form, where a not equal to zero has the below properties:

• It is the two-degree equation where the value of the polynomial can be determined by the multiplicity of 2
• The complex numbers form the coefficients with close fields and thereby the two roots are equal
• The sum of the roots is equal to -b/a and the product of the root is defined as the c/a

Explain the Process of Solving Quadratic Equations with Common Roots

The process of solving quadratic equations with two common roots can be determined by the multiplication of 2. The polynomial factors in the quadratic equation have repeated roots and it is defined as n1, n2 etc for the corresponding roots of a,b, and c. Therefore, the factored form of the polynomial can be written by:

(x-a)n1 (x-b)n2 (x-c)n3

Formulas of Quadratic Equation

The standard formula of the quadratic equation is ax2 + bx + c = 0, however, the quadratic equation can be defined and denotes by other forms:

a (x – h)2 + k = 0

The above equation is denoted as the vertex type of the quadratic equation.

a (x – p)(x – q) = 0

The above equation is defined as the intercept form of the quadratic equation. 

Conclusion

The above study indicates that the value of the coefficient in the quadratic equation can not be zero and the roots are equal to each other in the close fields. The standard form of the quadratic equation suggests that the value of the common roots can be determined by finding the relations between coefficient and roots. The above study indicates that the form of a polynomial in the common root is defined as (x-a) n for equal root a. Here in the quadratic equation of common roots, the value of n is 2.