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Chapter 10: Problem 30

Multiply \((2 y-5)^{2}\)

Short Answer

Expert verified
\((2y - 5)^2 = 4y^2 - 20y + 25\)

Step by step solution

01

Identify the values of a and b

To expand the expression \((2y-5)^2\), first identify the values of a and b in the formula (a - b)^2 = a^2 - 2ab + b^2. In our case, a = 2y and b = 5.
02

Expand the expression using the formula

Now, expand the expression \((2y - 5)^2\) using the formula (a - b)^2 = a^2 - 2ab + b^2. Substitute the values of a and b we found in Step 1. \((2y - 5)^2 = (2y)^2 - 2(2y)(5) + (5)^2\)
03

Simplify the expression

Simplify the expression obtained in Step 2 by performing the calculations. \((2y - 5)^2 = (4y^2) - (20y) + 25\)
04

Write the final result

After simplifying the expression, we get the final result: \((2y - 5)^2 = 4y^2 - 20y + 25\)

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Key Concepts

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

Understanding the Binomial Theorem
The binomial theorem is a powerful tool in algebra that helps us expand expressions raised to a power. When you see something like \((a - b)^2\), it might seem complicated at first. But with the binomial theorem, it becomes much clearer and easier to handle.

Key Formula:
For the expression \((a - b)^2\), the binomial theorem tells us that it expands to \(a^2 - 2ab + b^2\). This means:
  • Square the first term: \(a^2\)
  • Multiply the two terms together, double it, and negate it: \(-2ab\)
  • Square the last term: \(b^2\)
Using the formula makes expanding binomials much simpler and is a fundamental skill in algebra. Just identify \(a\) and \(b\) and use this rule to expand efficiently!
Exploring Quadratic Expressions
Quadratic expressions form a crucial part of algebra. They are polynomials of degree two, which means the highest exponent of the variable is 2. An expanded binomial \((2y - 5)^2\) becomes the quadratic \(4y^2 - 20y + 25\).

Components of Quadratic Expressions:
A typical quadratic expression \(ax^2 + bx + c\) has three parts:
  • The quadratic term \(ax^2\) (in our case, \(4y^2\))
  • The linear term \(bx\) (here, \(-20y\))
  • The constant term \(c\) (which is \(+25\))
Quadratics can describe a wide array of phenomena in different fields, from physics to finance. Recognizing their pattern allows you to predict their shape (a parabola) and solve equations more easily.
Mastering Mathematical Simplification
Mathematical simplification is the process of reducing expressions to their simplest form. After expanding an expression like \((2y - 5)^2\), simplification is the next logical step.

Steps for Simplification:
  • Perform all multiplications: \((2y)^2 = 4y^2\) and \(2(2y)(5) = 20y\)
  • Add like terms, if necessary (e.g., combining \(-20y\) with other similar terms if present)
  • Re-evaluate constants to simplify them (\(5^2 = 25\))
Simplification ensures that expressions are as concise and understandable as possible, making them easier to work with in future calculations or analyses.

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Most popular questions from this chapter

Rationalize the denominator of each expression. $$\frac{\sqrt{3}}{\sqrt{28}}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[5]{\frac{3}{8}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{\sqrt{u}}{\sqrt{u}-\sqrt{v}}$$

Simplify completely. Assume all variables represent positive real numbers. $$\sqrt[4]{t^{36}}$$

Rationalize the denominator of each expression. $$\frac{25}{\sqrt{10}}$$
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