Question

How do you factor \[6{{x}^{2}}+7x-49\]?

Answer 1

Hint: This type of problem is based on the concept of factoring a polynomial. First, we have to consider the whole polynomial with degree 2. We have to split the middle term of the quadratic polynomial in such a way that we get common terms from the first and last term. Here, the middle term is 7x. We can split the middle as an addition of -14x and 21 x. We have to take 2x common from the first two terms and 7 common from the last two terms. We find that (3x-7) is common in the two terms. On taking the common terms, we get the factors of the polynomial.

Complete step by step solution:
According to the question, we are asked to find the factors of \[6{{x}^{2}}+7x-49\].
We have been given the polynomial is \[6{{x}^{2}}+7x-49\]. -----(1)
The given polynomial is of degree 2 and variable x.
To find the factors of the polynomial, we have to consider the middle term and split the middle term in such a way that the addition of these terms will be 7 and multiplication will be \[6\times -49\].
We know that \[6\times -49=-249\]. Thus the addition should be 7 and multiplication of these two numbers should be -249.
We know that -14+21=7 and \[21\times -14=-249\].
Therefore, we get
\[6{{x}^{2}}+7x-49=6{{x}^{2}}+\left( -14+21 \right)x-49\]
Using the distributive property, that is \[a\left( b+c \right)=ab+ac\], we get
\[6{{x}^{2}}+7x-49=6{{x}^{2}}+\left( -14 \right)x+21x-49\]
\[\Rightarrow 6{{x}^{2}}+7x-49=6{{x}^{2}}-14x+21x-49\]
We can express the polynomial as
\[6{{x}^{2}}+7x-49=3\times 2{{x}^{2}}-7\times 2x+3\times 7x-{{7}^{2}}\]
We find that 2x are common in the first two terms of the simplified polynomial and 7 are common in the last two terms of the simplified polynomial.
Let us take 2x and 7 common out of the bracket respectively.
\[\Rightarrow 6{{x}^{2}}+7x-49=2x\left( 3x-7 \right)+7\left( 3x-7 \right)\]
We find that 3x-7 is common in both the terms of the equation. On taking (3x-7) common, we get
\[6{{x}^{2}}+7x-49=\left( 3x-7 \right)\left( 2x+7 \right)\]
Here, we find that the polynomial is converted as a product of two linear polynomials.
These are the factors of the given polynomial.
Therefore, the factors of \[6{{x}^{2}}+7x-49\] are 3x-7 and 2x+7.

Note: Whenever you get this type of problem, we should always try to make the necessary changes in the given equation to get the final solution of the equation which will be the required answer. We should avoid calculation mistakes based on sign conventions. If the polynomial is of degree 2, then we get two factors only.