Question

How do you factor completely $5-25x-30{{x}^{2}}$?

Answer 1

Hint: Now to factor the given expression we will use splitting the middle term method. To use this method we will split the middle terms such that the terms on multiplication will give the product of first term and last term. Now we will simplify the expression by taking -30x common from the first two terms and 5 common from last two terms. Hence on further simplification we get the factors of the given expression.

Complete step by step solution:
Now the given expression is a quadratic expression in x.
Now rearranging the terms of the given expression we get, $-30{{x}^{2}}-25x+5$
Now to factor the given expression we will use splitting the middle term method.
Now to use this method we will split the middle terms such that the multiplication of the two terms gives us the product of the first term and last term.
Now here the middle term is $-25x$ , the first term is $-30{{x}^{2}}$ and the last term is 5.
Now let us write $-25x$ as $-30x+5x$ in the expression.
Hence we get the given expression as,
$\Rightarrow -30{{x}^{2}}-30x+5x+5$
Now taking -30x common from the first two terms and 5 common from the last two terms we get,
$\Rightarrow -30x\left( x+1 \right)+5\left( x+1 \right)$
Now again taking $\left( x+1 \right)$ common from the whole expression we get,
$\Rightarrow \left( x+1 \right)\left( -30x+5 \right)$
Hence we get the factors of the expression are $\left( -30x+5 \right)$ and $\left( x+1 \right)$ .
Hence the given expression can be factored as $\left( -30x+5 \right)\left( x+1 \right)=5-25x-30{{x}^{2}}$ .

Note: Now note that the given expression is a quadratic expression. Now we can also skip this method and find the roots of the given expression. We can find the roots using complete square method. To use this method we will first divide the whole polynomial with a and then add the term ${{\left( \dfrac{b}{2a} \right)}^{2}}$ . Now we will simplify the expression using the formula $\left( a\pm b \right)={{a}^{2}}+{{b}^{2}}\pm 2ab$
Now we will separate the complete square and take the root to simplify the expression. Hence we get the roots of quadratic expression. Now note that if we get the roots as $\alpha $ and $\beta $ then the factors of the expression are $x-\alpha $ and $x-\beta $ .