Find Quadratic Equation Given Sum and Product of its Roots

Example on how to find a quadratic equations given the sum and product of its roots.

Example 1

Find the values of the coefficients $b$ and $c$ in the quadratic equation $2 x^2+b x+c$ if the sum and product of the roots of this equation are respectively $2\dfrac{1}{2}$ and $3\dfrac{1}{4}$.
  1. $b= 2\dfrac{1}{2} , c = 3\dfrac{1}{4}$

  2. $b= 6\dfrac{1}{2} , c = -5 $

  3. $b= 3\dfrac{1}{4} , c = 2\dfrac{1}{2} $

  4. $b= -5 , c = 6\dfrac{1}{2}$

  5. $b=-2 , c = -2$

Solution


  1. The sum $s$ and the product $p$ of the roots of a quadratic equation of the form $a x^2+b x+c$ are given by

    $s=\dfrac{-b}{a}$ and $p=\dfrac{c}{a}$

  2. The coefficient $a$ in the given equation is known. Hence

    $s=2\dfrac{1}{2}=\dfrac{-b}{2}$ and $p=3\dfrac{1}{4}=\dfrac{c}{2}$

  3. Convert the left side of each equation into a fraction and rewrite the above equations as follows

    $\dfrac{5}{2}=\dfrac{-b}{2}$ and $\dfrac{13}{4}=\dfrac{c}{2}$

  4. Solve for $b$ and $c$: $b=-5$ and $c=6\dfrac{1}{2}$

    Answer D