Question

Factor completely using trial and error-

$18x^2-3xy-10y^2$.

Short answer

The factors of$18x^2-3xy-10y^2$are$(6x-5y)(3x+2y)$.

Step-by-step solution

Step 1. Given and explanation

We have an equation-$18x^2-3xy-10y^2$. We will first find out all the factors of the first and the last term.
Then by trying them with each other, we will find the correct set of factors which will result into the given equation when multiplied.

Step 2. Finding out the factors of the first and last term.

The first term is $18$.

Factors of it are-role="math" localid="1644569242500" $(1\times 18),(2\times 9),(6\times 3)$

Last term is $-10$.

Factors of it will be-role="math" localid="1644569259093" $(1\times -10),(10\times -1),(2\times -5),(5\times -2)$

Step 3. Trying out the factors.

The possibilities are-

Possible factors	Product
$(x+y)(18x-10y)$	Not possible option
role="math" localid="1644569858167" $(x+10y)(18x-y)$	$18x^2+179xy-10y^2$
$(x+2y)(18x-5y)$	$18x^2+31xy-10y^2$
role="math" localid="1644569887310" $(x-5y)(18x+2y)$	Not possible option
$(2x+y)(9x-10y)$	$18x^2-11xy-10y^2$
role="math" localid="1644569919663" $(2x-10y)(9x+y)$	Not possible option
$(2x+2y)(9x-5y)$	Not possible option
role="math" localid="1644569947054" $(2x-5y)(9x+2y)$	$18x^2-41xy-10y^2$
$(6x+y)(3x-10y)$	$18x^2-57xy-10y^2$
role="math" localid="1644569965032" $(6x-10y)(3x+y)$	Not possible option
$(6x+2y)(3x-5y)$	Not possible option
$(6x-5y)(3x+2y)$	$18x^2-3xy-10y^2$

"Not possible options" are declared so, because as there is no common factor in the equation, there cannot be any common factor in any of the factors also.

Step 4. Final Answer and checking the solution.

As we can see from the above table, we can conclude that the correct factors are those whose product matches with the given equation.

So the correct factors are $(6x-5y)(3x+2y)$.

We will check the solution by multiplying. If we get the same given equation, our calculations are right.

So,
$$\begin{gathered} =(6x-5y)(3x+2y) \\ =18x^2+12xy-15xy-10y^2 \\ =18x^2-3xy-10y^2 \end{gathered}$$

Thus our calculations are right.