How do you factor the expression $10{x^2} - 3x - 1$?

Hint: Given a quadratic equation. We have to find the factors of the equation. First, multiply the coefficient of the leading term by the constant term and then factorize the equation by splitting the middle term whose sum is equal to middle term and product is equal to the product of first and last term. After that find some common terms and determine the factors of the expression.

Complete step by step solution:
We are given the equation, $10{x^2} - 3x - 1$.

Now, we will multiply the coefficient of ${x^2}$ and the constant term.

$ \Rightarrow \left( {10} \right)\left( { - 1} \right) = - 10$

Find the factors of $ - 10$, we get:

$ \Rightarrow - 10 = - 2 \times 5$

$ \Rightarrow - 10 = 2 \times - 5$

$ \Rightarrow - 10 = 1 \times - 10$

$ \Rightarrow - 10 = - 1 \times 10$

Now, determine the factor which will give $ - 3$ on adding.

$ \Rightarrow - 5 + 2 = - 3$

Therefore, required factor of middle term splitting is $2$ and $ - 5$

Now, write the expression by splitting the middle using the factors.

$ \Rightarrow 10{x^2} - 5x + 2x - 1$

Now, take $5x$ common from first two terms and 1 as common from the last two terms, we get:

$ \Rightarrow 5x\left( {2x - 1} \right) + 1\left( {2x - 1} \right)$

Now, we will take $\left( {2x - 1} \right)$ as a common factor from the above expression.

$ \Rightarrow \left( {2x - 1} \right)\left( {5x + 1} \right)$

Final answer: Hence the factors of the expression $10{x^2} - 3x - 1$ are $\left( {2x - 1} \right)$ and $\left( {5x + 1} \right)$

Note: Please note that the students can check the above factors by equating each bracket equal to zero and determine the value of x.

$ \Rightarrow \left( {2x - 1} \right) = 0$

$ \Rightarrow x = \dfrac{1}{2}$

Equating $\left( {5x + 1} \right)$ to zero, we get:

$ \Rightarrow \left( {5x + 1} \right) = 0$

$ \Rightarrow x = - \dfrac{1}{5}$

Now, substitute the value of $x = \dfrac{1}{2}$ in the above expression, we get:

$ \Rightarrow 10{\left( {\dfrac{1}{2}} \right)^2} - 3\left( {\dfrac{1}{2}} \right) - 1$
$ \Rightarrow 10\left( {\dfrac{1}{4}} \right) - 3\left( {\dfrac{1}{2}} \right) - 1$
$ \Rightarrow \dfrac{5}{2} - \dfrac{3}{2} - 1$
$ \Rightarrow \dfrac{{5 - 3 - 2}}{2} = 0$

Since, by substituting $x = \dfrac{1}{2}$ into the expression, the result is equal to zero, therefore, the factor $\left( {2x - 1} \right)$ is a factor of the expression.

 Now, we will also check $\left( {5x + 1} \right)$ is a factor of the expression by substituting $x = - \dfrac{1}{5}$ into the expression.

$ \Rightarrow 10{\left( { - \dfrac{1}{5}} \right)^2} - 3\left( { - \dfrac{1}{5}} \right) - 1$
$ \Rightarrow 10\left( {\dfrac{1}{{25}}} \right) + 3\left( {\dfrac{1}{5}} \right) - 1$
$ \Rightarrow \dfrac{2}{5} + \dfrac{3}{5} - 1$
$ \Rightarrow \dfrac{{2 + 3 - 5}}{5} = 0$

Since, by substituting $x = - \dfrac{1}{5}$ into the expression, the result is equal to zero, therefore, the factor $\left( {5x + 1} \right)$ is a factor of the expression.