Question

How do you factor \[2{{x}^{2}}+9x+7\]?

Answer 1

Hint: In this type of problem, we should have proper knowledge in the topic of polynomials. In this problem, first, we will find all the terms and coefficients of the equation. Then, we will multiply the quadratic term and the constant term to get the sum as a linear term. After that, we will find the pairs of the multiplied term. From there, we can find the factors by putting the pairs in the place of the linear terms.

Complete step by step answer:
In this example, the quadratic term is \[2{{x}^{2}}\] and the constant term is 7. Now, we will multiply them and write all the factors of the multiplied result in pairs.
Multiply of \[2{{x}^{2}}\] and 7 = \[14{{x}^{2}}\]
Factors of the result are:
-1x and -14x
1x and 14x
-2x and -7x
2x and 7x
From the above pairs, we have to find that adds to produce the linear term.
We want to pair +9x. For this problem, the answer is 2x and 7x because
\[2x\times 7x=14{{x}^{2}}\] and \[2x+7x=9x\]
Now, from the equation, break up the linear term into two terms using the numbers 2x and 7x, we get
\[2{{x}^{2}}+2x+7x+7\]
Always put a plus sign between two sets:
\[\left( 2{{x}^{2}}+2x \right)+\left( 7x+7 \right)\]
Look at the first two sets. They share something in common. They are sharing 2x. If you factor out the 2x, we will have 2x(x+1). Now, look at the second two terms. They are sharing a 7. If you factor out the 7, then we have 7(x+1). The polynomial is now written as 2x(x+1)+7(x+1). We can see a common term (x+1) in the equation 2x(x+1)+7(x+1).
Now, we can write the above equation as (2x+7)(x+1) because of the common factor (x+1).
Therefore, the factors of \[2{{x}^{2}}+9x+7\] is \[\left( 2x+7 \right)\left( x+1 \right)\]

Note:
 If someone follows all the above steps, then he/she will have the way to the solution. Be careful of the signs when someone does this. We can use a different method to do this problem. For that, we will have a proper knowledge in quadratic equation. By using the formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] in the equation \[2{{x}^{2}}+9x+7\] as a= 2, b =+9, and c= 7, we get
\[x=\dfrac{-9\pm \sqrt{81-4\times 2\times 7}}{2\times 2}\]
\[\Rightarrow x=\dfrac{-9\pm 5}{4}\]
\[\Rightarrow x=\dfrac{-7}{2}\text{ and x=}-1\]
Therefore, factors are \[\left( 2x+7 \right)\text{ and }\left( x+1 \right)\] .