Question

How do you factor $ 6{x^2} - 30x + 36? $

Answer 1

Hint: Take the given expression and take all the terms on one side of the equation and then find factors of the terms by splitting the middle terms and then simplify for the required values of “x”.

Complete step by step solution:
Take the given expression: $ 6{x^2} - 30x + 36 $
The above equation Can be re-written as –
 $ \Rightarrow 6{x^2} - 30x + 36 = 0 $
Take the common multiple from all the three given terms.
\[\Rightarrow 6({x^2} - 5x + 6) = 0\]
The above expression can be re-written as –
\[\Rightarrow {x^2} - 5x + 6 = 0\]
Now we will use the concept of to split the middle term.
Here we have three terms in the given expression.
Now, multiply the constant in the first term with the last term.
i.e. $ 1 \times 6 = 6 $
Now, you have to split the middle term to get $ ( - 6) $ in multiplication and addition or subtraction to get middle term i.e. $ ( - 5) $ . Here applying the basic concept of the product of two negative terms gives us the positive term and addition of two negative terms gives the value in the negative sign.
 $
 \Rightarrow 6 = ( - 2) \times ( - 3) \\
 \Rightarrow ( - 5) = ( - 2) + ( - 3) \\
  $
Write the equivalent value for the middle term –
\[{x^2} - 3x - 2x + 6 = 0\]
Now, make the pair of two terms in the above equation-
\[\underline {{x^2} - 3x} - \underline {2x + 6} = 0\]
Find the common factors from the paired terms –
\[\Rightarrow x(x - 3) - 2(x - 3) = 0\]
Take the common factors in the above equation –
 $ \Rightarrow (x - 3)(x - 2) = 0 $
 $ \Rightarrow x = 2,3 $

Hence, the solution of $ 6{x^2} - 30x + 36 $ is $ x = 2,3 $.

Note: Here we were able to split the middle term and find the factors but in case it is not possible then we can find factors by using the formula \[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\] and considering the general form of the quadratic equation $a{x^2} + bx + c = 0$ . Be careful about the sign convention and simplification of the terms in the equation.