What is $x^2+21 x+110$ in factored form?
Answer (2)
Find two numbers that multiply to 110 and add up to 21.
The numbers are 10 and 11.
Write the factored form using these numbers: ( x + 10 ) ( x + 11 ) .
The factored form of the quadratic expression is ( x + 10 ) ( x + 11 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression x 2 + 21 x + 110 . Factoring a quadratic means expressing it as a product of two binomials.
Finding the Numbers To factor the quadratic x 2 + 21 x + 110 , we need to find two numbers that multiply to 110 (the constant term) and add up to 21 (the coefficient of the x term).
Listing Factor Pairs Let's list the factor pairs of 110:
(1, 110) (2, 55) (5, 22) (10, 11)
Checking the Sum Now, let's 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) adds up to 21.
Writing the Factored Form Since the numbers 10 and 11 multiply to 110 and add up to 21, we can write the factored form of the quadratic as ( x + 10 ) ( x + 11 ) .
Final Answer Therefore, the factored form of x 2 + 21 x + 110 is ( x + 10 ) ( x + 11 ) .
Examples
Factoring quadratic expressions is useful in many real-world applications. For example, suppose you want to design a rectangular garden with an area represented by the quadratic expression x 2 + 21 x + 110 . By factoring this expression into ( x + 10 ) ( x + 11 ) , you determine that the dimensions of the garden can be ( x + 10 ) and ( x + 11 ) . This allows you to plan the layout of your garden efficiently. Factoring also helps in solving problems related to projectile motion, where the height of an object can be modeled by a quadratic equation.
The quadratic expression x 2 + 21 x + 110 can be factored into the form ( x + 10 ) ( x + 11 ) . This is found by identifying two numbers that multiply to 110 and add to 21, which are 10 and 11. Thus, the final factored form is ( x + 10 ) ( x + 11 ) .
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