What is $p^2-20 p+51$ in factored form?
Answer (2)
Find two numbers that multiply to 51 and add up to -20.
Identify the two numbers as -3 and -17.
Write the factored form using these numbers: ( p − 3 ) ( p − 17 ) .
The factored form of the quadratic expression is ( p − 3 ) ( p − 17 ) .
Explanation
Understanding the Problem We are given the quadratic expression p 2 − 20 p + 51 and we want to factor it. Factoring a quadratic means expressing it as a product of two binomials.
Finding the Numbers To factor the quadratic expression p 2 − 20 p + 51 , we need to find two numbers that multiply to 51 and add up to -20. Let's call these two numbers a and b . So, we are looking for a and b such that:
a b = 51 a + b = − 20
Determining the Signs Since the product a b is positive (51) and the sum a + b is negative (-20), both a and b must be negative numbers.
Listing the Factors Now, let's find the factors of 51. The factors of 51 are 1, 3, 17, and 51. We can express 51 as a product of two factors in the following ways:
1 × 51 = 51 3 × 17 = 51
Finding the Correct Pair Since both numbers must be negative, we can consider the negative factors:
− 1 × − 51 = 51 , but − 1 + ( − 51 ) = − 52 , which is not -20. − 3 × − 17 = 51 , and − 3 + ( − 17 ) = − 20 . This works!
Writing the Factored Form So, the two numbers we are looking for are -3 and -17. Therefore, the factored form of the quadratic expression is:
( p − 3 ) ( p − 17 )
Checking the Answer To check our answer, we can expand the factored form:
( p − 3 ) ( p − 17 ) = p 2 − 17 p − 3 p + 51 = p 2 − 20 p + 51
This matches the original quadratic expression, so our factored form is correct.
Final Answer The factored form of the quadratic expression p 2 − 20 p + 51 is ( p − 3 ) ( p − 17 ) .
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 of the garden is given by the expression p 2 − 20 p + 51 , where p is related to the dimensions of the garden. By factoring this expression into ( p − 3 ) ( p − 17 ) , you can determine the possible dimensions of the garden in terms of p . This allows you to plan the layout of your garden based on the desired area and constraints on the dimensions. Factoring helps in solving problems related to area, optimization, and design in various fields.
The quadratic expression p 2 − 20 p + 51 factors to ( p − 3 ) ( p − 17 ) . This is found by identifying two numbers that multiply to 51 and add to -20, which are -3 and -17. Verifying the factorization shows it matches the original expression.
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