Question

Find the value of $m$ so that the quadratic equation $m\left( {x - 7} \right) + 49 = 0$ has two equal roots.

Answer 1

Hint: For finding the value of $m$ for the equation we will use the concept for equal roots. As we know that since the quadratic equation has equal roots, hence its discriminant will be equal to zero. So by using this concept we will solve this problem.

Formula used:
If a quadratic equation $a{x^2} + bx + c$ is there in the equation, then its discriminant will be calculated by ${b^2} - 4ac$ .
Here, $a,b,c$ are the variables.

Complete step-by-step answer:
Since we have the equation given as $m\left( {x - 7} \right) + 49 = 0$ . And also it is given that it has two equal roots hence the discriminant will be equal to zero.
So on expanding the quadratic equation, we get
$ \Rightarrow m{x^2} - 7mc + 49 = 0$
From the above equation, we can see that the equation follows the standard quadratic equation hence, ${b^2} - 4ac = 0$ .
Now on substituting the values, we get the equation as
$ \Rightarrow {\left( { - 7m} \right)^2} - 4 \times 49 \times m = 0$
On solving the square and the multiplication of the above solution, we get
$ \Rightarrow 49{m^2} - 4 \times 49m = 0$
Taking the common factor common, we get
$ \Rightarrow 49m\left( {m - 4} \right) = 0$
And on solving it, we will get the two value of $m$ and it will be as
$ \Rightarrow m = 0,4$
Therefore, the value of $m$ will be equal to $0$ and $4$ .

Note: There is more than one way to solve the problem of the quadratic equation. Some of them are named as factoring and it is also known as splitting a middle term. In this, we don’t get the equation like that where we have to do such. Another one is completing the square. And the one which we had used in this is the quadratic formula. This is widely used when none of the above two works for an equation.