Question

Factorise $ {x^2} - 9x + 20 $

Answer 1

Hint: First of all take the given expression and then will use the method of splitting the middle term to get the factors of the given quadratic equation. We have to split in such a way that the middle term is the product of the term which is equal to the product $ 1 \times (20) = (20) $ t of the first and last term.

Complete step-by-step answer:
Take the given expression: $ {x^2} - 9x + 20 = 0 $
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.
Now, you have to split the middle term to get $ ( + 20) $ in multiplication and addition or subtraction to get the middle term i.e. $ ( - 9) $ . 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.
 $
  ( + 20) = ( - 4) \times ( - 5) \\
  ( - 9) = - 4 - 5 \;
  $
Write the equivalent value for the middle term –
 $ {x^2} - \underline {4x - 5x} + 20 = 0 $
Now, make the pair of two terms in the above equation-
 $ \underline {{x^2} - 4x} - 5\underline {x + 20} = 0 $
Find the common factors from the paired terms –
 $ x(x - 4) - 5(x - 4) = 0 $
Take the common factors in the above equation –
 $ (x - 4)(x - 5) = 0 $
Hence, factor of $ {x^2} - 9x + 20 $ is $ (x - 4)(x - 5) $
So, the correct answer is “ $ (x - 4)(x - 5) $ ”.

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