Question

How do you factor completely $3{{x}^{2}}-x-10$?

Answer 1

Hint: We will look at the product of the constant term and the leading coefficient of the given quadratic equation. Then we will look at pairs of factors of this product. We will find a pair of factors such that the sum or difference of these factors will result in the coefficient of the middle term. Then we will split the middle term and obtain the factorization of the given equation.

Complete answer:
The given quadratic equation is $f\left( x \right)=3{{x}^{2}}-x-10$. The leading coefficient is the coefficient of the ${{x}^{2}}$ term, which is 3 in this equation. The product of the leading coefficient and the constant term is the following,
$3\times -10=-30$
Let us look at some pairs of factors of the number 30. We know that $15\times 2=30$ and $10\times 3=30$. We also have $5\times 6=30$. Now, if we take the factors 5 and 6, then we can see that $-6+5=-1$ and $-6\times 5=-30$. Let us split the middle term in the following manner,
$f\left( x \right)=3{{x}^{2}}-6x+5x-10$
Now, we will take out $3x$ as a common factor from the first and the second term. And we will take out 5 as a common factor from the third and the fourth term. So, we get the following,
$f\left( x \right)=3x\left( x-2 \right)+5\left( x-2 \right)$
Next, we will take out $\left( x-2 \right)$ as a common factor. So, we have the following,
$f\left( x \right)=\left( x-2 \right)\left( 3x+5 \right)$
Thus, we have obtained the factors for the given quadratic equation.

Note: It is possible to further solve the quadratic equation after factorization. The product of two factors can be zero only when one of the factors is zero. We equate each of the factors to zero to find the possible values of the variable. So, the solutions for the given equation are $x=2$ and $x=-\dfrac{5}{3}$. There are other methods to solve the quadratic equations. These methods are the completing square method and the quadratic formula method.