Question

How do you factor $2{x^2} + 9x - 5$?

Answer 1

Hint: In this algebraic problem, we have given some quadratic expression. Here we are asked to find the factors of the given quadratic expression. For this we have found the factors of the coefficients of ${x^2}$ term and the constant term and whose products add to the coefficient of $x$. By this way we can find the factors.

Complete step-by-step solution:
Given quadratic term is $2{x^2} + 9x - 5$
We asked to factorize the given term.
First we are going to use the sum product pattern to solve the given quadratic term.
Now, using the sum product pattern method we are going to write the given term $2{x^2} + 9x - 5$ as $2{x^2} + 10x - x - 5 - - - - - (1)$.
Next we are going to take the common factor from the two pairs. Let us consider the first two terms of equation (1) as first pair and next two terms of equation (1) as second pair. Now, these two pairs take the common term out.
$ \Rightarrow 2{x^2} + 10x - x - 5 = 2x\left( {x + 5} \right) - 1\left( {x + 5} \right)$
Now, there are only two terms and every term is multiplied by $\left( {x + 5} \right)$. So $\left( {x + 5} \right)$ is common on both the terms.
Now, rewriting in factored form, we get
$ \Rightarrow 2x\left( {x + 5} \right) - 1\left( {x + 5} \right) = \left( {2x - 1} \right)\left( {x + 5} \right)$
Therefore we factorized the given term.

Hence,$\left( {2x - 1} \right)\left( {x + 5} \right)$ is the required solution.

Note: We factorized the given quadratic term. And given a quadratic containing the unknown variable $x$, we can also find the value of $x$ by equating the given term to zero.
Then we have to factorize the given quadratic term like we did above. Then by equating the factors to zero, we can find the value of $x$.
$ \Rightarrow \left( {2x - 1} \right)\left( {x + 5} \right) = 0$
$ \Rightarrow \left( {2x - 1} \right) = 0$And$\left( {x + 5} \right) = 0$
$ \Rightarrow x = \dfrac{1}{2}$ And $x = - 5$.