Question

Solve for x: \[15{x^2} - 7x - 36 = 0\]
A. \[\dfrac{5}{9}, - \dfrac{4}{3}\]
B. \[\dfrac{9}{5}, - \dfrac{4}{3}\]
C. \[\dfrac{9}{5}, - \dfrac{3}{4}\]
D. None

Answer 1

Hint: Start by analysing the coefficients in the equation and use the factoring method to find the factors of the equation. The factoring method is basically taking the coefficient of x and splitting it such that the coefficient of ${x^2}$ and constant when multiplied gives that number.

Complete Step-by-Step solution:
Here, the quadratic equation is \[15{x^2} - 7x - 36 = 0\].
As we can see, the middle term is having coefficients -7 and that of first and last are 15 and -36.
$ \Rightarrow 15 \times 36 = 540$. We can split the middle term with coefficients 27 and 20 because $27 \times 20$ is also 540 and 27 – 20 is 7. Doing so, we’ll get:
\[ \Rightarrow 15{x^2} - 27x + 20x - 36 = 0\]
Now taking $3x$ outside from first two terms and $4$ outside from the next two terms, we’ll get:
\[ \Rightarrow 3x\left( {5x - 9} \right) + 4\left( {5x - 9} \right) = 0\]
From here, we can take \[\left( {5x - 9} \right)\] outside from the entire equation, we’ll get:
\[
   \Rightarrow \left( {3x + 4} \right)\left( {5x - 9} \right) = 0 \\
   \Rightarrow \left( {3x + 4} \right) = 0{\text{ or }}\left( {5x - 9} \right) = 0 \\
   \Rightarrow x = - \dfrac{4}{3}{\text{ or }}x = \dfrac{9}{5} \\
\]
Therefore, the values of x in the above equation are \[ - \dfrac{4}{3}\] and \[\dfrac{9}{5}\]. B is the correct option.

Note: If we are getting any difficulty in finding the roots of a quadratic equation by factorisation, we can use a direct formula to find it. If $a{x^2} + bx + c = 0$ is a given quadratic equation then its roots are:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
If we use this, we’ll get the same result.