Factor.
[tex]x^2-15 x+50[/tex]
(x-10)(x-5)
(x+5)(x+10)
(x-10)(x+5)
(x-5)(x+10)
Answer (1)
To factor the quadratic expression:
Find two numbers that multiply to 50 and add up to -15.
The numbers are -5 and -10.
Write the factored form as ( x − 5 ) ( x − 10 ) .
The factored form is ( x − 10 ) ( x − 5 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression x 2 − 15 x + 50 . This means we want to find two binomials of the form ( x + a ) ( x + b ) such that when we multiply them, we get the original quadratic expression.
Finding the Factors To factor the quadratic expression x 2 − 15 x + 50 , we need to find two numbers that multiply to 50 and add up to -15. Let's list the pairs of factors of 50:
1 and 50 2 and 25 5 and 10
Since we need the two numbers to add up to -15, we can consider the negative factors:
-1 and -50 -2 and -25 -5 and -10
We see that -5 and -10 add up to -15 and multiply to 50.
Writing the Factored Form Therefore, the factored form of the quadratic expression is ( x − 5 ) ( x − 10 ) .
Checking the Answer We can check our answer by expanding the factored form:
( x − 5 ) ( x − 10 ) = x 2 − 10 x − 5 x + 50 = x 2 − 15 x + 50
This matches the original quadratic expression, so our factorization is correct.
Final Answer The factored form of the quadratic expression x 2 − 15 x + 50 is ( x − 10 ) ( x − 5 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, if you are designing a rectangular garden with an area of x 2 − 15 x + 50 square feet, you can use factoring to determine the possible dimensions of the garden. In this case, the dimensions could be ( x − 10 ) feet and ( x − 5 ) feet. Understanding factoring allows you to solve problems related to area, optimization, and other practical scenarios.
