Problem 78 - Elementary Algebra

Question

Solve by completing the square. \(n^{2}-2 n=-3\)

Step-by-Step Solution

Answer

The solutions are \( n = 3 \) and \( n = -1 \).

Step 1: Move the constant to the right side

Add 3 to both sides of the equation to move the constant term to the right side: \[ n^2 - 2n + 3 = 0 \]

Step 2: Prepare to complete the square

Rearrange the equation to make it easier to complete the square: \[ n^2 - 2n = 3 \]

Step 3: Add and subtract the square of half the coefficient of n

Take half of the coefficient of \( n \) (which is -2), square it (\( (-1)^2 = 1 \)), and add it to both sides: \[ n^2 - 2n + 1 = 3 + 1 \]This simplifies to: \[ n^2 - 2n + 1 = 4 \]

Step 4: Write the left side as a perfect square

Write the left side of the equation as a square of a binomial: \[ (n - 1)^2 = 4 \]

Step 5: Solve for n

Take the square root of both sides: \[ n - 1 = \text{±} 2 \]Then solve for \( n \): \[ n - 1 = 2 \] or \[ n - 1 = -2 \]This gives two solutions: \[ n = 3 \] and \[ n = -1 \]

Key Concepts

solving quadratic equations

Quadratic equations are equations of the form \text{ax}^2 + bx + c = 0\text{ where a, b\text{,} and c are constants. Solving these types of equations means finding the values of \(x\) that satisfy the equation. Completing the square is one effective method to solve quadratic equations. It involves manipulating the equation to form a perfect square trinomial on one side. This allows us to easily solve for \(x\) using the square root property.

algebraic manipulations

Algebraic manipulations are fundamental to solving quadratic equations through completing the square. Here's a breakdown:

• First, move the constant term to the right side of the equation.
• Next, take half of the coefficient of the \(x\) term, square it, and add it to both sides of the equation. This ensures we form a perfect square trinomial on the left side.
• Finally, rewrite the trinomial as a square of a binomial, which simplifies solving by using the square root property.

These steps involve adding, subtracting, and rearranging terms, ensuring the equation transforms correctly.

perfect square trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It has the form: \text{ax}^2 + 2abx + b^2 = (ax + b)^2 .
In the exercise, we transformed \(n^2 - 2n + 1\) into \((n - 1)^2\). This was achieved by

• Taking half of the coefficient of \(n\), which is \(-2\), giving us \(-1\).
• Squaring \(-1\) to get 1, and adding this to both sides of the equation.

This manipulation turns \(n^2 - 2n + 1\) into \((n - 1)^2\), making it easier to solve.

square root property

The square root property is a useful tool when we have an equation in the form of \((x - k)^2 = p\).
It states that:
x - k = ±√p. Applying this to our example \((n - 1)^2 = 4\text{,} we,
Take the square root of both sides: n - 1 = ±2. Simplify the equation to find: n = 3 \text{,} and, n = -1. This gives us two potential solutions for \)n$, showing the versatility and simplicity of the square root property.