Question

How do you factor the expression $6{{x}^{2}}-23x+15$?

Answer 1

Hint: As the given equation is a quadratic equation so we will use the splitting middle term method to solve the equation. In this method we will split the middle term of the equation $6{{x}^{2}}-23x+15$ such as the product of two numbers is equal to $15\times 6$ and the sum of two numbers is equal to $23$.

Complete step by step solution:
We have been given an equation $6{{x}^{2}}-23x+15$.
We have to find the factors of the given equation.
$\Rightarrow 6{{x}^{2}}-23x+15$
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 $a\times c=6\times 15=90$ and their sum is equal to $b=23$.
So we will use two numbers as 18 and 5.
So splitting the middle term we will get
$\Rightarrow 6{{x}^{2}}-\left( 18x+5x \right)+15$
Now, simplifying the above obtained equation we will get
$\Rightarrow 6{{x}^{2}}-18x-5x+15$
Now, taking the common terms out we will get
$\Rightarrow 6x\left( x-3 \right)-5\left( x-3 \right)$
Now, again taking common factors out we will get
$\Rightarrow \left( 6x-5 \right)\left( x-3 \right)$
Hence we get the factors of the given equation as $\left( 6x-5 \right)\left( x-3 \right)$.

Note: To solve a quadratic equation splitting the middle term method is the easiest way. Alternatively we can use completing the square method to solve the equation. For this we have to add or subtract a number from the equation such that the constant term becomes a perfect square. Then we will simplify the LHS of the obtained equation and solve the equation for x.