Relationships between Coefficient and Roots of a Quadratic Equation

Coefficients in a quadratic equation contain effective information regarding the product and the sum of its roots. There are multiple types of quadratic equations such as standard form: y=ax2+bx+c, factored form: y = (ax + c)(bx + d), and vertex form: y=a(x+b)2+c. The article is going to focus on the relationship between coefficient and roots as well as quadratic equations. In mathematics, a quadratic equation can be defined as a second-degree equation and the form is ax² + bx + c = 0.

Quadratic Equation

In mathematics, quadratic equations are polynomial equations with the form of ax² + bx + c = 0.  In this case, x is a variable, c is a constant term, and a and b are coefficients. The root of quadratic equation through the quadratic formula is:

(, ) = [–b √(b2 – 4ac)]/2a

The quadratic equations primarily have two roots and the root quadratic equation can be either imaginary or real. On the subject of the root quadratic equation, there are mainly two values of “x”, that are achieved through quadratic equation solving. The root of the equation is referred to through the symbols of beta (β) and alpha (α). The root of the equation ax² + bx + c = 0 is nothing rather than the solution of quadratic equations. There are different methods that are involved in the roots of an equation such as factoring, graphing, completing the square as well as the quadratic formula.

The root of the equation can be defined as the variables’ values that effectively satisfy the equation. These values are also well known as the “zeros” or “solutions” of quadratic equations. For instance:

x2 – 7x + 10 = 0

In this case, x = 2 as well as x = 5 as they effectively satisfy the equation. 

There are mainly three kinds of roots such as real and equal roots, real and distinct roots,  as well as complex roots. The word “quadratic” is obtained from the term “Quad” and the meaning of the term is square. Moreover, it can be stated in other words that the quadratic equation is “equation of degree 2”. 

Relationships between Coefficient and Roots

In the formula of quadratic equations ax² + bx + c = 0, a,b, c are coefficients and x is an unknown variable. Therefore, this equation cannot be effectively called a quadratic equation and the figures of a,b, and c are also known as quadratic coefficients. The sum of quadratic equation’s roots is effectively equal to second term coefficient divided through the leading coefficient. For instance, describing the relationship between coefficient and roots:

In case the difference between roots of equation x2 – 13x + k = 0  is 17 find k. 

Solution:

x2-13x + k =0 here, a =1, b = -13, c = k

Consider, α, β are the roots of equation,

Thus, +=(-b)/a=(–13))/1=13        (1)

Thus – = 17     (2)

Thus 2α=30provides α=15

Therefore, 15 + = 13 (from (1))

However, αβ = c/a = k/1 provides 15 (-2) = k

Thus, k = -30

The roots of an equation can be calculated effectively through the formula of  x = (-b √ (b² – 4ac) )/2a. The roots of an equation can be defined as a value that is effectively substituted in an equation for the unknown quantity that satisfies the equation. There is an effective significance of the root of the equation as it assists in dividing the x-axis.  

Conclusion 

The article discussed the relationship between coefficient and roots of quadratic equations.  The root of the equation in quadratic equations calculated with the formula of x = (-b √ (b² – 4ac) )/2a. The sum of the roots of the quadratic equation is -b/a and their product is c/a.