Question

If the product of the roots of the equation \[(a + 1){x^2} + (2a + 3)x + (3a + 4) = 0\] is $2$, then sum the roots.

Answer 1

Hint:First we will assume the roots of the given equation as $\alpha $ and $\beta $. Then using the formula for the product of roots of the quadratic equation we will find the value of $a$. After finding the value of $a$ we will put the value in the formula for the sum of roots of the quadratic equation to get the required solution.

Formula Used:
The general form of the quadratic equation: $ax^2+bx+c=0$
Sum of roots of quadratic equation $ = \dfrac{{ - b}}{a}$
Product of roots of quadratic equation $ = \dfrac{c}{a}$

Complete step by step Solution:
Given, equation is \[(a + 1){x^2} + (2a + 3)x + (3a + 4) = 0\]
Let the $\alpha $ and $\beta $ are the roots of the equation \[(a + 1){x^2} + (2a + 3)x + (3a + 4) = 0\]
$\alpha + \beta = \dfrac{{ - (2a + 3)}}{{a + 1}}$ (1)
$\alpha \beta = \dfrac{{3a + 4}}{{a + 1}}$
Given, $\alpha \beta = 2$
$\dfrac{{3a + 4}}{{a + 1}} = 2$
Cross multiplying the above equation
$3a + 4 = 2(a + 1)$
After simplifying the above equation
$3a + 4 = 2a + 2$
Shifting variable to one side and constant to the other side
$3a - 2a = 2 - 4$
After solving the above equation
$a = - 2$
Putting in equation (1)
$\alpha + \beta = \dfrac{{ - (2(-2) + 3)}}{{ - 2 + 1}}$
$\alpha + \beta = \dfrac{{ - (-4 + 3)}}{{ - 2 + 1}}$
After simplifying the above expression
$\alpha + \beta = \dfrac{{ 1}}{{ - 1}}$
After solving the above expression
$\alpha + \beta = -1$
Hence, the sum of the roots is $-1$

Additional Information:The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. It also goes by the name quadratic equations. The quadratic equation has the following generic form:

where a, b, and c are numerical coefficients and x is an unknown variable. Here, a is greater than zero because if it equals zero, the equation will cease to be quadratic and change to a linear equation.
Sum of roots of quadratic equation $ = \dfrac{{ - b}}{a}$
Product of roots of quadratic equation $ = \dfrac{c}{a}$

Note: Students should know the concept for solving the question before start solving the question. They should use formula correctly that Sum of roots of quadratic equation $ = \dfrac{{ - b}}{a}$ not $\dfrac{b}{a}$. Many students can do that mistakes and they will get an incorrect answer. So, should take care of that.