Question

How do you factor completely $6{{x}^{2}}+19x+10$ ?

Answer 1

Hint: Now to factor the given expression we will use the splitting the middle term method. In this method we will split the middle term such that the product of the two terms is the multiplication of the first term and the last term. Further we will simplify the expression by grouping common terms from the first two terms and last two terms of the expression obtained.

Complete step by step solution:
Now we are given with a quadratic expression in x.
We want to factor the whole expression.
Factors are the expression which can divide the given expression without leaving a remainder.
Now we want to write the expression in terms of its factor.
To factorize the expression we will use splitting the middle term method.
Now consider the given expression $6{{x}^{2}}+19x+10$
The middle terms of the expression is 19x.
Now we will split the middle term such that the product of the two terms is the multiplication of first term and last term.
Hence we can split 19x as 15x + 4x as we have $\left( 15x \right)\left( 4x \right)=60{{x}^{2}}=10\times 6{{x}^{2}}$ hence we get,
$\Rightarrow 6{{x}^{2}}+15x+4x+10$
Now taking 3x common from the first two terms and 2x common from the last two terms we get,
$\Rightarrow 3x\left( x+5 \right)+2\left( x+5 \right)$
Now taking $\left( x+5 \right)$ common from the above expression we get,
$\Rightarrow \left( 3x+2 \right)\left( x+5 \right)$
Hence we get the factor of the given expression.
Hence we have $\left( x+5 \right)\left( 3x+2 \right)=6{{x}^{2}}+19x+10$

Note: Now note that the given expression is quadratic of the form $a{{x}^{2}}+bx+c$ . Now we can directly write the factors of the quadratic as we know the roots of quadratic are given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we have the factors of the quadratic a $\left( x-\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \right)$