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Chapter 5: Problem 71

Solve each equation for \(y .\) Assume a and b are positive numbers. $$y^{2}+b y=0$$

Short Answer

Expert verified
The solutions are y = 0 and y = -b .

Step by step solution

01

Understand the Equation

The given equation is y^2 + by = 0 . This is a quadratic equation in the standard form of ax^2 + bx + c = 0 .
02

Factor the Quadratic Equation

To solve for y , factor out the common term y from the given equation: y(y + b) = 0 .
03

Apply the Zero Product Property

According to the Zero Product Property, if ab = 0 , then a = 0 or b = 0 . Apply this property to the factored equation: y = 0 or y + b = 0 .
04

Solve for y from Each Equation

Solve the equations obtained from the Zero Product Property: y = 0 is already in its simplest form. For y + b = 0 , subtract b from both sides to isolate y : y = -b .
05

Compile the Solutions

The solutions to the equation y^2 + by = 0 are y = 0 and y = -b .

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Key Concepts

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

Zero Product Property

The Zero Product Property is a useful tool in algebra, especially when dealing with factored quadratic equations. This property states that if a product of two terms is zero, then at least one of the terms must be zero.

  • Simply put, if ab = 0, then either a = 0 or b = 0 (or both).

Isolating Variables
Isolating variables is a fundamental algebraic technique used to solve for unknowns. After applying the Zero Product Property, we end up with simpler equations like y = 0 and y + b = 0. To isolate the variable y in the second equation, consider the following steps:
  • y + b = 0
  • Subtract b from both sides: y = -b.

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