2.3 Relationship Between the Roots and Coefficients of a Quadratic Equation

2.3 Relationship Between the Roots and Coefficients of a Quadratic Equation

The standard form of a quadratic equation is $ax^2 + bx + c = 0, a \neq 0$ and $a, b, c \in R$.

The roots of this equation are:

$\dfrac{-b + \sqrt{b^2 - 4ac}}{2a}$ and $\dfrac{-b - \sqrt{b^2 - 4ac}}{2a}$

Let these roots be $\alpha, \beta$

$\alpha = \dfrac{-b + \sqrt{b^2 - 4ac}}{2a}, \beta = \dfrac{-b - \sqrt{b^2 - 4ac}}{2a}$

Now $S =$ Sum of the roots $= \alpha + \beta$

$S =$ Sum of the roots $= -\dfrac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$

$P =$ Product of the roots $= \alpha\beta$

$P =$ Product of the roots $= \dfrac{c}{a} = \dfrac{\text{Constant term}}{\text{Coefficient of } x^2}$

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Important Note!
(i) The standard form of a quadratic equation is $ax^2 + bx + c = 0$ where $a \neq 0$ and $a, b, c \in R$.
(ii) Sum of the roots $= \dfrac{-b}{a}$, Product of the roots $= \dfrac{c}{a}$
(iii) When the coefficient of $x^2$ in a quadratic equation is unity (one).

• Sum of roots is equal to the coefficient of $x$ with sign changed.
• Product of the roots is equal to the constant term.
  (iv) If the constant term of $ax^2 + bx + c = 0$ is zero, then sum of its roots $= \dfrac{-b}{a}$ and products of the roots $= 0$.

Example 9

Find the sum and the product of the roots of the equation $3x^2 + 5x - 12 = 0$ without solving.

Solution Standard form of the quadratic equation is $ax^2 + bx + c = 0$, where $a \neq 0$ and $a, b, c \in R$. The given equation is $3x^2 + 5x - 12 = 0$.

Here $a = 3, b = 5, c = -12$

$S =$ Sum of the roots $= -\dfrac{b}{a} = -\dfrac{5}{3}$

$P =$ Product of the roots $= \dfrac{c}{a} = \dfrac{-12}{3} = -4$

Example 10

Find the sum and the product of the roots of the quadratic equation $x^2 + 6x + 8 = 0$

Solution Coefficient of $x^2$ is unity.

Sum of the roots $= -6$ [Coefficient of $x$ with sign changed]

Product of the roots $= 8$ [Constant term]