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 numbers as -3 and -17.
Write the factored form as ( 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 asked to factor it. Factoring a quadratic means expressing it as a product of two binomials.
Finding the Right Numbers We need to find two numbers that multiply to 51 (the constant term) and add up to -20 (the coefficient of the p term). Let's call these numbers a and b . So, we want to find a and b such that:
a b = 51 and a + b = − 20
Considering Factors of 51 The factors of 51 are 1, 3, 17, and 51. Since we need the two numbers to add up to a negative number (-20), we should consider the negative factors of 51, which are -1, -3, -17, and -51. We are looking for a pair of these factors that multiply to 51 and add to -20.
Identifying the Correct Pair We can see that − 3 × − 17 = 51 and − 3 + ( − 17 ) = − 20 . So, the two numbers we are looking for are -3 and -17.
Writing the Factored Form Now we can write the factored form of the quadratic expression using these two numbers:
p 2 − 20 p + 51 = ( p − 3 ) ( p − 17 )
Verification To verify our factorization, 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 factorization is correct.
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 must be p 2 − 20 p + 51 square feet. By factoring this expression into ( p − 3 ) ( p − 17 ) , you determine that the dimensions of the garden could be ( p − 3 ) feet by ( p − 17 ) feet. This allows you to plan the layout of your garden based on the desired area.
The expression p 2 − 20 p + 51 can be factored into ( p − 3 ) ( p − 17 ) . This is achieved by finding two numbers that multiply to 51 and add to -20, which are -3 and -17. Therefore, the factored form of the quadratic expression is ( p − 3 ) ( p − 17 ) .
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