Question

How do you factor $ \dfrac{{3{x^2} + 4x - 15}}{{2{x^2} + 3x - 9}} $ ?

Answer 1

Hint: First of all we will find the factors of the numerator and the denominator and then will place one upon the another and then will remove if there are any common multiple in the numerator and the denominator.

Complete step-by-step answer:
Numerator: $ 3{x^2} + 4x - 15 $
Here, use the concept of splitting 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. $ 3 \times ( - 15) = ( - 45) $
Now, you have to split the middle term to get $ ( - 45) $ in multiplication and addition or subtraction to get the middle term i.e. $ ( + 4) $ . 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.
 $
  ( - 45) = (9) \times ( - 5) \\
  ( + 4) = 9 - 5 \;
  $ $ $ $ $
Write the equivalent value for the middle term –
 $ = 3{x^2} + 9x - 5x - 15 $
Now, make the pair of two terms in the above equation-
 $ = \underline {3{x^2} + 9x} - \underline {5x - 15} $
Find the common factors from the paired terms –
 $ = 3x(x + 3) - 5(x + 3) $
Take the common factors in the above equation –
 $ = (x + 3)(3x - 5) $
Hence, the factor of $ 3{x^2} + 4x - 15 $ is $ (x + 3)(3x - 5) $ …. (A)
Similarly, Denominator: $ 2{x^2} + 3x - 9 $
Find the factors:
 $
   = 2{x^2} + 3x - 9 \\
   = 2{x^2} + 6x - 3x - 9 \;
  $
Make pair of two terms-
 $ = \underline {2{x^2} + 6x} - \underline {3x - 9} $
Take common multiples –
 $
   = 2x(x + 3) - 3(x + 3) \\
   = (x + 3)(2x - 3) \;
  $
Hence, the factors of $ 2{x^2} + 3x - 9 $ are $ (x + 3)(2x - 3) $ ….. (B)
By using the equations (A) and (B), place the values in the given expression –
 $ \dfrac{{3{x^2} + 4x - 15}}{{2{x^2} + 3x - 9}} = \dfrac{{(x + 3)(3x - 5)}}{{(x + 3)(2x - 3)}} $
Common factors from the numerator and the denominator cancel each other.
 $ \dfrac{{3{x^2} + 4x - 15}}{{2{x^2} + 3x - 9}} = \dfrac{{(3x - 5)}}{{(2x - 3)}} $
This is the required solution.
So, the correct answer is “$\dfrac{{(3x - 5)}}{{(2x - 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.