Quadratic polynomial:
The polynomial is an expression of degree \(n\) for the variable \(x\) is of the form \(p(x) = a_0 x^n + a_{1}x^{n-1} + … + a_{n -1}x + a_n\), where \(a_0, a_1, a_2, …, a_n\) are coefficients and \(a_0 \ne 0\).
A quadratic polynomial is a polynomial of degree \(2\) for the variable \(x\) is of the form \(p(x) = ax^2 + bx + c\), \(a \ne 0\) and \(a\), \(b\) and \(c\) are real numbers.
Zero of a quadratic polynomial:
If the value of the polynomial \(p(x)\) at \(x = k\) is zero \(p(k) = 0\), then the real number \(k\) is called the zero of the polynomial \(p(x)\).
Quadratic equation:
A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are numbers, \(a \ne 0\). The degree of the quadratic equation is \(2\).
Roots of a quadratic equation:
The value of \(x\) that makes the expression \(ax^2 + bx + c\) is zero, called the roots of the quadratic equation.
Relation between roots and coefficients of a quadratic equation:
If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\), then:
(i) Sum of the roots \(=\)
(ii) Product of the roots \(=\)
Formation of a quadratic equation:
If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, then the general formula to construct the quadratic equation is \(x^2 - (\alpha + \beta) x + \alpha \beta = 0\).
That is, \( x^2 - (\text{sum of roots}) x + \text{product of roots} = 0\).
Sum of the roots \(=\) \(\alpha + \beta\) \(=\) \(\frac{-b}{a}\)
Product of the roots \(=\) \(\alpha \beta\) \(=\) \(\frac{c}{a}\)
The solution of the quadratic equation is the value of the variable that makes the equation zero. We can say these solutions as roots or zeroes.