Question

Expand the following using identities $\left( {x + 45} \right)\left( {x - 20} \right)$

Answer 1

Hint: We will first write $\left( {x + 45} \right)\left( {x - 20} \right)$ as \[\left( {x + 45} \right)\left( {x + \left( { - 20} \right)} \right)\]. We will open the brackets using $\left( {a + b} \right)\left( {c + d} \right) = ac + ad + bc + bd$. Then, we will simplify the brackets using the rules of multiplication of integers. And at last adding like terms, will give us the final answer.

Complete step-by-step answer:
We have to expand the given polynomial $\left( {x + 45} \right)\left( {x - 20} \right)$ using identities
We can rewrite is as \[\left( {x + 45} \right)\left( {x + \left( { - 20} \right)} \right)\]
We know that $\left( {a + b} \right)\left( {c + d} \right) = ac + ad + bc + bd$
Hence, we will have $\left( {x + 45} \right)\left( {x - 20} \right)$ equivalent to
${x^2} + \left( { - 20} \right)x + 45x + 45\left( { - 20} \right)$
Now, the rule states that if a negative and positive quantity is multiplied, we get a negative answer, that is $ + \left( - \right) = - $.
Hence, we can write the expression as
=${x^2} - 20x + 45x - \left( {45} \right)\left( {20} \right)$
On simplifying the brackets, we will get,
=${x^2} - 20x + 45x - 900$
Now, we will add like terms.
=${x^2} + 25x - 900$
Hence, $\left( {x + 45} \right)\left( {x - 20} \right) = {x^2} + 25x - 900$.

Note: When two positive quantities are multiplied or two negative quantities are multiplied, the resultant is always positive whereas when a negative and a positive quantity is multiplied, the resultant is a negative quantity.