Question

Simplify the product \[\left( {x + 5} \right)\left( {x + 4} \right)\] and write it in standard form.

Answer 1

Hint: Multiply each individual term in the left bracket with each individual term in the right bracket and add them. This law has a specific name which is distributive law.

Complete step by step solution:
To simplify any product of the form \[\left( {a + b} \right)\left( {c + d} \right)\] multiply each term in the left parenthesis with each individual term in the right parenthesis and add them. In simple words multiply \[a\] with \[c\], then add it to the product of \[a\] and \[d\], then add this to the product of \[b\] and \[c\], and finally add it to the product of \[b\] and \[d\].
\[\left( {a + b} \right)\left( {c + d} \right)\] \[ = \] \[ac + ad + bc + bd\].
In the expression \[\left( {x + 5} \right)\left( {x + 4} \right)\], \[a = x\], \[b = 5\], \[c = x\], \[d = 4\]. Follow the above rule and multiply:
\[\left( {x + 5} \right)\left( {x + 4} \right)\] \[ = \] \[{x^2} + 4x + 5x + 20\]
Add the terms with same power of \[x\]:
\[ \Rightarrow \] \[\left( {x + 5} \right)\left( {x + 4} \right)\] \[ = \] \[{x^2} + 9x + 20\]
Standard form of writing any expression is arranging them in the decreasing power of the variable. In this case the variable is \[x\], the highest power of \[x\] is \[2\], and so the first term will be \[{x^2}\], then \[9x\], in which coefficient of \[x\] is \[1\], and then \[20\] in which power of \[x\] is \[0\].

Hence the simplified solution of the product in the standard form is \[{x^2} + 9x + 20\].

Note:
Students must be careful while multiplying the terms and maintain the correct order. After multiplying the terms with the same power of the variables must be added (or subtracted according to the question) in the final answer. Students must also know that the expression we obtained as the answer in this solution is known as a quadratic equation.