Question

Write the quadratic equation if the addition of roots is 10 and the product of the root is 9.

Answer 1

Hint: For this let us assume a quadratic equation $a{x^2} + bx + c = 0$and $\alpha \& \beta $are the roots of the equation. So, sum of the roots is $\alpha + \beta $ and product of the roots is $\alpha \beta $.
Formula for sum of roots and product of roots is given as:
$
   \Rightarrow \alpha + \beta = \dfrac{{ - b}}{a} \\
   \Rightarrow \alpha \beta = \dfrac{c}{a} \\
 $

Complete step by step solution: First of all, let us see what is given to us. The sum and product of the roots is given to us and we have to find quadratic equation.
For this, let us assume a quadratic equation given as follow:
$ \Rightarrow a{x^2} + bx + c = 0$ …………(1)
How we can find the quadratic equation? We can find it by applying the formula of the sum of roots and product of roots and compare it with values given to us.
It is given that sum of roots is equal to 10. Therefore,
$ \Rightarrow \alpha + \beta = 10$ …………(2)
$ \Rightarrow \alpha + \beta = \dfrac{{ - b}}{a}$ …………(3)
From (2) & (3)
$
   \Rightarrow \dfrac{{ - b}}{a} = 10 \\
   \Rightarrow \dfrac{{ - b}}{a} \Rightarrow \dfrac{{10}}{1} \\
   \Rightarrow - b = 10 \\
   \Rightarrow b = - 10\& a = 1 \\
 $
The product of the roots is also given to us i.e. 9. Therefore,
$ \Rightarrow \alpha \beta = 9$ …………..(4)
$ \Rightarrow \alpha \beta = \dfrac{c}{a}$ …………..(5)
From (4) & (5)

$
   \Rightarrow \dfrac{c}{a} = 9 \\
   \Rightarrow \dfrac{c}{a} \Rightarrow \dfrac{9}{1} \\
   \Rightarrow c = 9\& a = 1 \\
 $
By putting the values of a, b and c in the equation (1), we get
$ \Rightarrow a{x^2} + bx + c = 0$
$ \Rightarrow {x^2} - 10x + 9 = 0$

So, required quadratic equation is ${x^2} - 10x + 9 = 0$

Note: We can check our answer by finding the roots of this equation.
$
   \Rightarrow {x^2} - 10x + 9c = 0 \\
   \Rightarrow {x^2} - 9x - x + 9 = 0 \\
   \Rightarrow x(x - 9) - 1(x - 9) = 0 \\
   \Rightarrow (x - 1)(x - 9) = 0 \\
   \Rightarrow x = 1\& x = 9 \\
 $

We have found the roots now we can find the sum and product of the roots as,

$
   \Rightarrow sum = \alpha + \beta = 1 + 9 = 10 \\
   \Rightarrow product = \alpha \beta = 1 \times 9 = 9 \\
 $
Here we can see the sum and roots are same as given in the question. Hence it is verified that our answer is correct.