Question

Factors of \[\left( 42-x-{{x}^{2}} \right)\] are.

Answer 1

Hint: In order to find the factors of a quadratic equation\[\left( 42-x-{{x}^{2}} \right)\], we split the middle term \[\left( -x \right)\] that is with the variable \[x\] into two parts such that their sum equals the middle term itself and the product equals the multiplication of first\[\left( -{{x}^{2}} \right)\] and the third term\[\left( 42 \right)\].

Formula used:
The formula used here is ‘splitting the middle term method’. In this method, if we consider a quadratic equation,
\[a{{x}^{2}}+bx+c\]
In order to factorize the considered equation, We just split \[bx\] into the two parts, which are\[dx\] and \[ex\] such that ,
\[d+e=b \\
\Rightarrow de=ac\]
And then we factor out the common terms.

Complete step by step solution:
First of all, we start off by writing the given equation as
\[42-x-{{x}^{2}}\]
Now, in order to solve the equation, we use the splitting the middle term method, so, splitting the middle term \[\left( -x \right)\]into two parts,
\[42-7x+6x-{{x}^{2}}\]
This is the equation that is obtained after splitting the middle term,
And now after simplifying the above equation, we get
\[\Rightarrow 7\left( 6-x \right)+x\left( 6-x \right) \\
\therefore \left( 7+x \right)\left( 6-x \right) \\ \]
So , the factors of the equation \[\left( 42-x-{{x}^{2}} \right)\] are found to be \[\left( 7+x \right)\left( 6-x \right)\].

Note: Alternatively, we can also use the quadratic formula in order to solve the question i.e. for finding the factors of the equation \[\left( 42-x-{{x}^{2}} \right)\].
According to the quadratic formula ,
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
By this we get the two different values of \[x\].And the values \[{{x}_{+}}\] and \[{{x}_{-}}\] that can be calculated by taking plus and minus signs from\[\pm \], respectively, and the factors will be
\[a\left( x-{{x}_{+}} \right)\left( x-{{x}_{-}} \right)\]
Now solving for \[\left( 42-x-{{x}^{2}} \right)\]
\[x=\dfrac{-\left( -1 \right)\pm \sqrt{1-4\left( -1 \right)42}}{2\left( -1 \right)} \\
\Rightarrow x=\dfrac{1\pm \sqrt{1+4\times 42}}{-2} \\
\Rightarrow x=\dfrac{1\pm \sqrt{169}}{-2} \\
\Rightarrow x=\dfrac{1\pm 13}{-2} \\
\Rightarrow {{x}_{+}}=-7 \\
\Rightarrow {{x}_{-}}=6\]
i.e. factors are
\[\left( -1 \right)\left( x+7 \right)\left( x-6 \right) \\
\therefore \left( x+7 \right)\left( 6-x \right)\]