Question

How do you factor \[-{{x}^{2}}-2x+63\]?

Answer 1

Hint: First of all take -1 common from all the terms such that the coefficient of ${{x}^{2}}$ become positive. Now, use the middle term split method to factorize \[{{x}^{2}}+2x-63\]. Split 2x into two terms in such a way that their sum equals 2x and product equals \[-63{{x}^{2}}\]. For this process, find the prime factors of 63 and combine them in such a manner so that our condition is satisfied. Finally, take the common terms together and write \[-{{x}^{2}}-2x+63\] as a product of two terms given as (x – m) (x – n). Here, ‘m’ and ‘n’ are called zeroes of the polynomial.

Complete step by step solution:
Here, we have been asked to factorize the quadratic polynomial: \[-{{x}^{2}}-2x+63\]. First let us make the coefficient of ${{x}^{2}}$ positive. So, taking -1 common from all the terms, we get,
\[\Rightarrow -{{x}^{2}}-2x+63=-1\left( {{x}^{2}}+2x-63 \right)\]
Now, let us use the middle term split method for the factorization of \[{{x}^{2}}+2x-63\]. We need to split the middle term 2x into two terms such that their sum equals 2x and product equals \[-63{{x}^{2}}\]. For this we need to find the prime factors of 63 and group them in a suitable manner such that both the conditions may get satisfied.
We know that 63 can be written as: - \[63=3\times 3\times 7\] as the product of its primes. Now, we have to group these factors such that the conditions of the middle term split method are satisfied. So, we have,
(i) \[\left( 9x \right)+\left( -7x \right)=2x\]
(ii) \[\left( 9x \right)\times \left( -7x \right)=-63{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow -{{x}^{2}}-2x+63=-1\left( {{x}^{2}}+9x-7x-63 \right) \\
 & \Rightarrow -{{x}^{2}}-2x+63=-1\left( x\left( x+9 \right)-7\left( x+9 \right) \right) \\
\end{align}\]
Grouping the common terms together we have,
\[\begin{align}
  & \Rightarrow -{{x}^{2}}-2x+63=-1\left( x+9 \right)\left( x-7 \right) \\
 & \Rightarrow -{{x}^{2}}-2x+63=-\left( x+9 \right)\left( x-7 \right) \\
\end{align}\]
Hence, \[-\left( x+9 \right)\left( x-7 \right)\] is the factored form of the given quadratic polynomial.

Note: You may note that it was not necessary to take -1 common from all the terms at the initial step of the solution. You can also factor the expression with doing that. In such a case by using the middle term split method we will get (x + 9) (7 – x) as the factored form which is the same as the result we have obtained. You can also use completing the square method to get the required factored form. You can also use completing the square method to get the required factored form but it will be a bit lengthier process.