Chapter 7.7, Problem 12OE

To determine

Find the quadratic equation whose roots are $1\text{ and 9}$ .

Answer to Problem 12OE

The quadratic equation is $x^2-10x+9=0$ .

Explanation of Solution

Given information: It is given that the roots of the quadratic a equation are $1\text{ and 9}$ .

Formula used: If $\left(S\right)$ is the sum of roots of a quadratic equation in variable $\left(x\right)$ and $\left(P\right)$ is the product of its roots, then quadratic equation in variable $\left(x\right)$ is written as,

  $x^2-Sx+P=\mathrm{0....}\left(1\right)$

Calculation: Sine the roots of the quadratic equation are $1\text{ and 9}$ . Therefore evaluate sum and product of roots for the quadratic equation.

  $\begin{array}{c}\text{Sum of roots}\left(S\right)=\left(1\right)+\left(9\right)\\ =10\\ \text{Product of roots}\left(P\right)=\left(1\right)\left(9\right)\\ =9\end{array}$

Now, with the help of formula (1) substitute the evaluated sum and product of roots in (1) and evaluated the quadratic.

  $\begin{array}{c}{x}^2-Sx+P=\mathrm{0....}\left(\text{From }\left(1\right)\right)\\ x^2-\left(10\right)x+\left(9\right)=0\\ x^2-10x+9=0\end{array}$