Question

How do you factor $14{{k}^{2}}-9k-18$?

Answer 1

Hint: We will factor the given quadratic equation by using the splitting the middle term method. We will split the middle term of the equation $a{{x}^{2}}+bx+c=0$ such that the product of two numbers is equal to $a\times c$ and sum of two numbers is equal to $b$.

Complete step-by-step answer:
We have been given an equation $14{{k}^{2}}-9k-18$.
We have to write the given equation in factored form.
Now, we will use the split middle term method. We have to find two numbers such as the product of two numbers is equal to $a\times c=14\times 18=252$ and their sum is equal to $b=9$.
So we will use two numbers as 12 and 21.
So splitting the middle term we will get
\[\Rightarrow 14{{k}^{2}}-\left( 21k-12k \right)-18\]
Now, simplifying the above obtained equation we will get
$\Rightarrow 14{{k}^{2}}-21k+12k-18$
Now, taking the common terms out we will get
$\Rightarrow 7k\left( 2k-3 \right)+6\left( 2k-3 \right)$
Now, again taking common factors out we will get
$\Rightarrow \left( 7k+6 \right)\left( 2k-3 \right)$
Hence we get the factors of the given equation as $\left( 7k+6 \right)\left( 2k-3 \right)$.

Note: Here in this question we use the split middle term method as it is a simple question. If possible make the coefficient of a to 1 by dividing the equation by some suitable number. We can also use other methods like quadratic formula, completing the square method also to solve the quadratic equations. Also we can find the values of k by equating each factor to zero and by solving the obtained equations.