 The equation ax square plus m equal to negative 3x has two distinct roots. The sum of the roots is 4 by 7. Find the value of a where a is the coefficient of x square. Now as we have seen in our earlier videos of this unit, for a standard quadratic equation ax square plus bx plus c equal to 0. Let's say if the roots are alpha and beta. So for the standard quadratic equation, the sum of the roots that is alpha plus beta would be given as negative of coefficient of x that is negative of b in this case divided by coefficient of x square which is a. And similarly the product of roots alpha times beta, this would be given as constant value c, constant value c divided by coefficient of x square that is a. And if you might question how did we get these values? So there are multiple ways to get to this. Either you can use quadratic formula and just write down the expression for roots. For example if the roots are alpha and beta according to quadratic formula, the roots will be negative b plus square root of discriminant divided by 2a and negative b minus of square root of discriminant divided by 2a. So when you would add them positive square root of d and negative square root of d would cancel out each other negative b and negative b would give us negative 2 times b divided by 2a. 2 into would cancel out, so we would be left with negative b by a which is indeed the sum of roots. Similarly when you will multiply them, you will get the product of the roots as c by a. So having said that once we know what the relation between roots and the coefficients of a standard quadratic equation is, let us now try to solve this question. So in the question it is given that the sum of the roots is 4 by 7 which means the value alpha plus beta is 4 by 7. But make sure your quadratic equation is in standard form. In this case we have ax square plus m and this is equal to negative 3x. So we need all the terms on one side of the equation. So in order to do that we would add 3x on both the sides. So plus 3x and plus 3x. So this and this would cancel out. So we would be left with ax square plus 3x plus m and this is equal to 0. So now our quadratic equation is in standard form. So for this quadratic equation the sum of the roots would be given as negative of coefficient of x that is negative of 3 divided by coefficient of x square that is a that is a. So negative 3 upon a this would be the sum of roots alpha plus beta. But sum of the roots is given as 4 by 7 which means 4 by 7 is equal to negative 3 by a. Now on multiplying both sides with a on multiplying both sides with a now this and this would cancel out and 4 a by 7 would be equal to negative 3 and now on multiplying both sides with a reciprocal of 4 by 7 that is 7 by 4 7 by 4 4 by 7 would cancel out 7 by 4 and we would be left with a equals to negative 7 3's are 21 divided by 4. So this is the value of a that we were looking for. The value of a that would satisfy the given equation that is the coefficient of x square would be equal to negative 21 divided by 4. She would get old by 2 years.