Algebraic Method of Solving Quadratic Equations

Solving the quadratic equation in an algebraic formula requires putting all of the variables and terms on one side of the equal sign and zero on the other side of the equal sign. The algebraic method refers to the use of the direct formula of factoring and grouping rather than the formula of the root function determination in quadratic maths. As for example if the general quadratic equation is ax2 + bx + c= 0 and the discriminant b2-4ac is in positive value then the formula of the quadratic equation will be x=-b+/√b2-4ac/2a or x= -b-√b2-4ac/2a.

What is a Quadratic Equation?

A quadratic equation is the algebraic equation of the second degree in the coefficient of x. The quadratic equation is defined as the equation that has degree 2. The general formula of the quadratic equation is denoted as ax2 + bx + c= 0, where a and b are the coefficient, x is the variable and c is the constant. The value of the coefficient is equal to each other and cannot be zero in the common factor equation. The roots of the quadratic formula are denoted by the below equation:

[-b ± √(b² – 4ac)]/2a

The discriminant of the above equation is defined as b2-4ac and if the discriminant is greater than zero then the roots are real and distinct.

What is the Algebraic Form of a Quadratic Equation?

The algebraic form of the quadratic equation can be denoted as ax2 + bx + c= a(x-r) (x-s)=0. Here r and s are the solutions of x determined by the formula of grouping or factoring. Determination of the square roots in the quadratic equation helps to get the result of the equation. The quadratic equation contains the power of x that is a positive integer. Therefore, the quadratic equation is also termed the polynomial algebraic equation. In the polynomial quadratic equation, the greatest number of the power of coefficient is 2. 

Solving of Quadratic Equation

Let the quadratic equation is 2x2 +4x -4=0 

x2 +2x -2=0

x2 +2x=2

x2 +2x+1=2+1

(x+1)2= 3

x+1= + √3 and – √3

x= 1 + or – √3

The plus and minus sign indicates that if the value of x is positive then the graph will be in the upward direction and if the value of x is negative then the graph slope will be a downward direction.

Algebraic Formula List

The list of the algebraic formula is given below:

• a2 – b2 = (a – b)(a + b)
• (a + b)2 = a2 + 2ab + b2
• a2 + b2 = (a + b)2 – 2ab
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
• a4 – b4 = (a – b)(a + b)(a2 + b2)
• an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1) (n is natural number in the equation)

What is Basic Algebra?

The basics of algebra refer to the numbers, factors or integers of the equation to find the value of the constant in the given equation. For example, 3x+6= 30 is the basic equation of algebra. Therefore, x is denoted as the variable in the equation, plus sign is the operator, 3 is denoted as the coefficient and 6 and 30 is the constant in the equation. The linear equations of the algebra are denoted as the below formula:

ax+b=c, ax+by+c=0 and ax+by+cz+d=0.

Properties of the Algebraic Equation

The properties of algebra for addition and multiplication in the quadratic equation state that 0 is denoted as the additive identity and 1 is defined as the multiplicative identity. In the case of the inverse relation, -a is denoted as the additive inverse and 1/a is the multiplicative inverse.

The cumulative properties of the algebraic equation is denoted as a+b=b+a, ab=ba. The associative properties of algebra are defined as a+(b+c)= (a+b)+c, a(bc)=(ab)c.

Conclusion

The above study indicates that the value of x in the algebraic equation can be positive or negative in terms of the factors in the equation. However, the value of the coefficient depends on the properties of the algebraic formula of addition or multiplication in the equation. The distributive property of the algebraic equation is denoted by 2(2x-3) = 4x-6. The factors in the algebraic equation can be determined by the formula of the square root equation. The inverse relation of the roots is due to the equal nature between coefficient and constant.