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Chapter 1: Problem 102

Solve the equation. $$ (y-3)^{2}=(y+1)^{2} $$

Short Answer

Expert verified
y = 1

Step by step solution

01

Expand Both Sides

Expand both sides of the equation \( (y-3)^2 = (y+1)^2 \). We obtain: \[ y^2 - 6y + 9 = y^2 + 2y + 1 \]
02

Move All Terms to One Side

Move all the terms to one side of the equation to set it to zero: \[ y^2 - 6y + 9 - y^2 - 2y - 1 = 0 \]. Simplify to get: \[ -8y + 8 = 0 \]
03

Simplify the Equation

Divide both sides by -8 to solve for \( y \): \[ y = 1 \]
04

Verify the Solution

Plug \( y = 1 \) back into the original equation to verify: \[ (1-3)^2 = (1+1)^2 \] which simplifies to \[ (-2)^2 = 2^2 \] or \[ 4 = 4 \]. This confirms that \( y = 1 \) is a solution.

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Key Concepts

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

quadratic equations
A quadratic equation is any equation that can be written in the form:
\[ ax^2 + bx + c = 0 \]
Here,
  • \( a \) is the coefficient of \( x^2 \)
  • \( b \) is the coefficient of \( x \)
  • \( c \) is the constant term
Each quadratic equation forms a parabola when graphed. In the exercise, the equation \( (y-3)^2 = (y+1)^2 \) is a quadratic equation set in a factored form.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying equations to find the values of unknowns. In this problem, the goal was to solve for \( y \).
Here are some steps taken in the solution:
  • First, expand \( (y-3)^2 \) and \( (y+1)^2 \) to get: \( y^2 - 6y + 9 \) and \( y^2 + 2y + 1 \), respectively.
  • Next, move all terms to one side of the equation: \( y^2 - 6y + 9 - y^2 - 2y - 1 = 0 \). Simplify this to: \( -8y + 8 = 0 \).
  • Finally, divide both sides by -8, giving: \( y = 1 \).
verification of solutions
Verifying solutions in algebra is a crucial step to ensure that the values found are correct.
In this case, substitute \( y = 1 \) back into the original equation:
  • Original equation: \( (1-3)^2 = (1+1)^2 \)
  • Simplify both sides: \( (-2)^2 = 2^2 \)
  • You get: \( 4 = 4 \)
Since both sides are equal, \( y = 1 \) is indeed a correct solution. Verification confirms that the solution fits the original equation, ensuring accuracy.

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