Factorise $a^2+14 a+45$ fully
Answer (1)
Find two numbers that multiply to 45 and add to 14.
Identify the numbers as 5 and 9.
Write the factorization as ( a + 5 ) ( a + 9 ) .
The fully factorized form is ( a + 5 ) ( a + 9 ) .
Explanation
Understanding the Problem We are given the quadratic expression a 2 + 14 a + 45 and our goal is to factorize it completely. This means we want to express it as a product of two binomials of the form ( a + x ) ( a + y ) , where x and y are constants.
Finding the Right Numbers To factorize the quadratic expression a 2 + 14 a + 45 , we need to find two numbers, x and y , such that their product is equal to the constant term (45) and their sum is equal to the coefficient of the linear term (14). In other words, we need to find x and y such that:
x × y = 45 and x + y = 14
Identifying the Factors We need to find two numbers that multiply to 45 and add up to 14. Let's list the factor pairs of 45:
1 and 45 (sum is 46)
3 and 15 (sum is 18)
5 and 9 (sum is 14)
We see that the pair 5 and 9 satisfy both conditions: 5 × 9 = 45 and 5 + 9 = 14 .
Writing the Factorization Now that we have found the two numbers, 5 and 9, we can write the factorization of the quadratic expression as:
a 2 + 14 a + 45 = ( a + 5 ) ( a + 9 )
Final Answer Therefore, the fully factorized form of the given quadratic expression is ( a + 5 ) ( a + 9 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you want to design a rectangular garden with an area of a 2 + 14 a + 45 square feet. By factoring this expression into ( a + 5 ) ( a + 9 ) , you determine that the dimensions of the garden could be ( a + 5 ) feet and ( a + 9 ) feet. This allows you to plan the layout of your garden based on the value of a .
