What is $x^2+21 x+110$ in factored form?
Answer (2)
We need to factor the quadratic expression x 2 + 21 x + 110 .
Find two numbers that multiply to 110 and add up to 21. These numbers are 10 and 11.
Write the factored form as ( x + 10 ) ( x + 11 ) .
The factored form of the quadratic expression is ( x + 10 ) ( x + 11 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 21 x + 110 and we want to factor it. Factoring a quadratic expression means rewriting it as a product of two binomials. In general, we want to find two numbers that add up to the coefficient of the x term (which is 21) and multiply to the constant term (which is 110).
Finding the Factors Let's find two numbers a and b such that a + b = 21 and a × b = 110 . We can list the factor pairs of 110: (1, 110), (2, 55), (5, 22), (10, 11).
Identifying the Correct Pair Now, we check which of these pairs adds up to 21. We see that 10 + 11 = 21 . So, the numbers we are looking for are 10 and 11.
Writing the Factored Form Therefore, we can write the quadratic expression as ( x + 10 ) ( x + 11 ) . To verify this, we can expand the factored form: ( x + 10 ) ( x + 11 ) = x 2 + 11 x + 10 x + 110 = x 2 + 21 x + 110
Final Answer Thus, the factored form of x 2 + 21 x + 110 is ( x + 10 ) ( x + 11 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures, ensuring stability and optimal material usage. Computer scientists apply factoring in cryptography to secure data transmission. Financial analysts use factoring to model and predict market trends, helping investors make informed decisions. Factoring helps simplify complex problems into manageable components, enabling professionals across various fields to optimize processes and achieve desired outcomes.
The expression x 2 + 21 x + 110 can be factored into ( x + 10 ) ( x + 11 ) by finding two numbers that add up to 21 and multiply to 110. The correct pair of numbers is 10 and 11. When expanded, this factored form returns to the original quadratic expression.
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