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Chapter 5: Problem 83

Factor. $$ 9 x^{2}-30 x y+25 y^{2} $$

Short Answer

Expert verified
(3x - 5y)^2

Step by step solution

01

Identify the Quadratic Formula

Recognize that the given expression is a quadratic in the form
02

Rearrange and Group Terms

Rewrite the quadratic as Split the middle term to make it easier to factor. Set up the expression as
03

Factor by Grouping

Factor each group of terms. Observe that involves common factors:
04

Combine the Factors

Combine the factors from previous step:
05

Write the Final Factored Form

Write down the final factored form of the given quadratic expression:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic expression
A quadratic expression is a polynomial of degree two. It can be written in the general form: \(ax^2 + bx + c\) where \(a, b, \text{ and } c\) are constants. In the context of our example, \(9x^2 - 30xy + 25y^2\) is a quadratic expression in two variables, \(x\) and \(y\). Identifying the quadratic expression helps us use specific factoring techniques to simplify it.
factoring by grouping
Factoring by grouping involves rearranging a polynomial so that we can factor pairs of terms separately. This technique is especially useful when dealing with quadratics.

For our quadratic \(9x^2 - 30xy + 25y^2\), we start by grouping terms:
1. Group the first two terms and the last two terms: \((9x^2 - 30xy) + (25y^2)\).
2. Factor out the greatest common factor in each group.
3. This helps us see the common binomial factor.
common factors
A common factor is a term that appears in each part of an expression. In the example \(9x^2 - 30xy + 25y^2\), we can look for common factors in the groups:
1. For \(9x^2 - 30xy\), the common factor is\(3x\): \(3x(3x - 10y)\).
2. For \(25y^2\), the common factor is \(5y\): \(5y(5y - 10y)\).

This reveals the common binomial \((3x - 5y)\) in the factorization.
factored form
The factored form of a quadratic expression is a product of its factors. Converting the expression into this form simplifies multiplication and division. In our example \(9x^2 - 30xy + 25y^2\), the factored form is \((3x - 5y)(3x - 5y)\) or \((3x - 5y)^2\).

Factoring makes it easier to solve quadratic equations, analyze functions, and simplifies algebraic expressions.

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Most popular questions from this chapter

During the first 13 sec of a jump, a skydiver falls approximately \(11.12 t^{2}\) feet in \(t\) seconds. A small heavy object (with less wind resistance) falls about \(15.4 t^{2}\) feet in \(t\) seconds. Suppose that a skydiver jumps from \(30,000 \mathrm{ft},\) and \(1 \mathrm{sec}\) later a camera falls out of the airplane. How long will it take the camera to catch up to the skydiver?

For \(P(x)\) and \(Q(x)\) as given, find the following. $$ \begin{aligned} &P(x)=13 x^{5}-22 x^{4}-36 x^{3}+40 x^{2}-16 x+75\\\ &Q(x)=42 x^{5}-37 x^{4}+50 x^{3}-28 x^{2}+34 x+100 \end{aligned} $$ $$ 3[P(x)]-Q(x) $$

Factor completely. $$ 3(a+2)^{2}+30(a+2)+75 $$

Solve. Three consecutive even integers are such that the square of the third is 76 more than the square of the second. Find the three integers.

Solve. If each side of a square is lengthened by \(6 \mathrm{cm},\) the area becomes \(144 \mathrm{cm}^{2} .\) Find the length of a side of the original square.
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