What is $x^2+21 x+110$ in factored form?
Answer (2)
Find two numbers that multiply to 110 and add up to 21.
Identify the factor pairs of 110 and check their sums.
Determine that 10 and 11 satisfy both conditions: 10 × 11 = 110 and 10 + 11 = 21 .
Express the quadratic in factored form using these numbers: ( 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 means expressing it as a product of two binomials, in the form ( x + a ) ( x + b ) , where a and b are constants.
Finding the Numbers To factor the quadratic x 2 + 21 x + 110 , we need to find two numbers a and b such that their product is equal to the constant term (110) and their sum is equal to the coefficient of the x term (21). In other words, we need to find a and b such that:
a × b = 110
a + b = 21
Identifying the Correct Pair We can list the pairs of factors of 110:
(1, 110) (2, 55) (5, 22) (10, 11)
Now, we check which of these pairs adds up to 21:
1 + 110 = 111 2 + 55 = 57 5 + 22 = 27 10 + 11 = 21
The pair (10, 11) satisfies the condition a + b = 21 .
Writing the Factored Form Since we found that a = 10 and b = 11 (or vice versa), we can write the factored form of the quadratic expression as ( x + 10 ) ( x + 11 ) .
Verification To verify our answer, we can expand the factored form:
( x + 10 ) ( x + 11 ) = x 2 + 11 x + 10 x + 110 = x 2 + 21 x + 110
This matches the original quadratic expression, so our factored form is correct.
Final Answer Therefore, 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 has many real-world applications. For example, suppose you are designing a rectangular garden and you know the area is given by the expression x 2 + 21 x + 110 square feet. By factoring this expression into ( x + 10 ) ( x + 11 ) , you determine that the dimensions of the garden could be ( x + 10 ) feet and ( x + 11 ) feet. This allows you to plan the layout of your garden based on the value of x .
The factored form of the quadratic expression x 2 + 21 x + 110 is ( x + 10 ) ( x + 11 ) . This is derived by finding two numbers that multiply to 110 and add up to 21, specifically 10 and 11. Verification through expansion confirms the correctness of this factorization.
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