Question

Add the following expression: $3{{x}^{2}}+5x-4$ and $8x-2{{x}^{2}}+11$.
A. ${{x}^{2}}+13x+7$
B. ${{y}^{2}}+3x-7$
C. ${{x}^{2}}+13x+6$
D. ${{y}^{2}}-2x+7$

Answer 1

Hint: In this question, we should know that when we have to add two or more than two quadratic expressions, then only like terms are to be added together. For example, the term with the coefficient of ${{x}^{2}}$ in one expression can only be added to the term containing the coefficient of ${{x}^{2}}$ in the other expressions.

Complete step-by-step solution -
In this question, we have been asked to add two expressions, which are, $3{{x}^{2}}+5x-4$ and $8x-2{{x}^{2}}+11$. We know that when we have to add two quadratic expressions, we can add only the like terms, that is, we can add a term with the coefficient of $x$ in one expression with only a term with the coefficient of $x$ in the other expression. Here, we have to add the two expressions, $3{{x}^{2}}+5x-4$ and $8x-2{{x}^{2}}+11$. So, we can write it as,
$\left[ 3{{x}^{2}}+5x-4 \right]+\left[ 8x-2{{x}^{2}}+11 \right]$
We can also write it as,
$\left[ 3{{x}^{2}}+5x-4 \right]+\left[ -2{{x}^{2}}+8x+11 \right]$
Now, we know that we can add only the like terms. So, we will add all the terms with the coefficient of ${{x}^{2}}$, then we will add the terms with the coefficient of $x$ and then all the constant terms will also be added. So, we can write it as,
$\left[ 3{{x}^{2}}-2{{x}^{2}} \right]+\left[ 5x+8x \right]+\left[ -4+11 \right]$
Now, we can take the common terms outside. So, we will get,
$\begin{align}
  & \left( 3-2 \right){{x}^{2}}+\left( 5+8 \right)x+\left( -4+11 \right) \\
 & \Rightarrow {{x}^{2}}+13x+7 \\
\end{align}$
Hence, we get the sum of $3{{x}^{2}}+5x-4$ and $8x-2{{x}^{2}}+11$ as $x{{~}^{2}}+13x+7$.
Therefore, option (A) is the correct answer.

Note: In this question, the possible mistakes that the students can make is, by not writing $8x-2{{x}^{2}}+11$ as $\left[ -2{{x}^{2}}+8x+11 \right]$ and, because of this, at the time of adding the two expressions, that is, $3{{x}^{2}}+5x-4$ and $8x-2{{x}^{2}}+11$, they may make the mistake of adding $3x{{~}^{2}}$ and $8x$, which is wrong.