Question

How do you factor $2{x^2} + 11x + 5$ ?

Answer 1

In this question, we have to find the factor of a given equation, so we must know the definition of the factor and the type of the equation. The equation given in the question is quadratic as it has a degree equal to 2; a quadratic equation is a kind of polynomial equation.
The roots of an equation are defined as the values of the unknown variable at which the function comes out to be zero. As the value of y is zero at the x-axis, so they are simply the x-intercepts.
The factors of a quadratic equation can be found easily with the help of several methods like factorization, completing the square, quadratic formula, etc. Using the appropriate method, we can find out the correct answer.

Complete step by step answer:
The given equation is $2{x^2} + 11x + 5 = 0$
It can be solved by factorization as well as –
$2{x^2} + 11x + 5 = 0 \\\ \Rightarrow 2{x^2} + 10x + x + 5 = 0 \\\ \Rightarrow 2x(x + 5) + 1(x + 5) = 0 \\\ \Rightarrow (2x + 1)(x + 5) = 0 \\\ \Rightarrow 2x + 1 = 0,\,x + 5 = 0 \\\ \Rightarrow x = - \dfrac{1}{2},\,x = - 5 \\\$
Hence, the factors of the equation are $x + \dfrac{1}{2} = 0$ and $x + 3 = 0$ .

Note: We first convert the given equation in the standard equation form that is $a{x^2} + bx + c = 0$ for finding the roots by factorization. And then we compare the given equation and the standard equation and get the values of a, b and c.
Then we write b as a sum of two numbers such that their product is equal to the product of a and c, that is, ${b_1} \times {b_2} = a \times c$ . We find the value of ${b_1}$ and ${b_2}$ by hit and trial, if we are not able to solve an equation by factorization then we move to the quadratic formula.