Problem 62 - A Graphical Approach to College Algebra

Question

Solve each quadratic equation by completing the square. $$x^{2}+2 x=4$$

Step-by-Step Solution

Answer

The solutions are \(x = \sqrt{5} - 1\) and \(x = -\sqrt{5} - 1\).

Step 1: Move Constant to the Other Side

The initial equation is \(x^2 + 2x = 4\). To complete the square, we first need to move the constant term to the other side: \(x^2 + 2x - 4 = 0\) becomes \(x^2 + 2x = 4\). It's easier for the next steps to keep this format.

Step 2: Add and Subtract the Square Term

To complete the square, identify \(b\) from the expression \(x^2 + 2x\), which is 2. Divide it by 2 to get 1, and then square it to get 1. Add and subtract this square inside the equation: \(x^2 + 2x + 1 - 1 = 4\). The equation now transforms to \((x^2 + 2x + 1) - 1 = 4\).

Step 3: Express as a Perfect Square

The expression \(x^2 + 2x + 1\) is a perfect square trinomial and can be rewritten as \((x+1)^2\). Now we have: \((x+1)^2 - 1 = 4\).

Step 4: Solve for the Perfect Square

Add 1 to both sides to isolate the perfect square: \((x+1)^2 - 1 + 1 = 4 + 1\). This simplifies to \((x+1)^2 = 5\).

Step 5: Apply the Square Root Property

Take the square root of both sides: \(x+1 = \pm \sqrt{5}\). This gives two potential solutions for \(x\): \(x+1 = \sqrt{5}\) and \(x+1 = -\sqrt{5}\).

Step 6: Solve for x

Subtract 1 from each of the equations found in the previous step to isolate \(x\): \(x = \sqrt{5} - 1\) and \(x = -\sqrt{5} - 1\). These are the solutions to the original equation.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. It's particularly useful when the quadratic doesn't factor easily or when simplifying equations is necessary. The basic idea is to transform the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. This technique involves several key steps:

• First, ensure the equation is in the form of a standard quadratic, generally written as \( ax^2 + bx + c = 0 \).
• Next, move any constant term to the opposite side of the equation to make the math simpler.
• Identify the linear coefficient (b), divide it by 2, and then square the result. This creates the term that transforms the existing expression into a perfect square trinomial.
• Add and subtract this number inside the equation, effectively rearranging without changing its value.

Completing the square rearranges the quadratic into a more manageable form, allowing us to easily find the variable's value.

Perfect Square Trinomial

A perfect square trinomial is a type of quadratic expression that can be rewritten as a binomial squared. It simplifies an expression of the form \( (ax^2 + bx + c) \) into \( (x + n)^2 \). Here's how it works:

• You start with your quadratic, for example, \( x^2 + 2x \). The goal is to add a constant to create a perfect square trinomial.
• The magic number is found by taking half of the linear coefficient (\( b \)) and squaring it. If you have \( b = 2 \), half of 2 is 1, and 1 squared is still 1.
• This transforms \( x^2 + 2x \) into \( x^2 + 2x + 1 \), i.e., \( (x+1)^2 \).

By converting into a perfect square trinomial, the quadratic becomes an expression that is easier to solve, especially when applying the square root property.

Square Root Property

The square root property is an essential tool for solving equations that have been converted into a perfect square trinomial. Once a quadratic equation is simplified to the form \((x+n)^2 = k\), the square root property allows you to solve for the variable without more complex algebraic manipulations:

• Take the square root of both sides of the equation. This is possible because the square root and squaring are inverse operations.
• Remember to include both the positive and negative roots when solving, since squaring any number (positive or negative) results in a positive value. For \((x+1)^2 = 5\), this means \( x+1 = \pm \sqrt{5} \).
• This provides you with two potential solutions: \( x+1 = \sqrt{5} \) and \( x+1 = -\sqrt{5} \).

The square root property simplifies the solving process, allowing quick identification of potential solutions.

Solving Quadratic Equations

Solving quadratic equations using the method of completing the square is systematic and involves organizing the equation to simplify finding the roots. After converting an equation into a perfect square trinomial, the steps to isolate and find the variable are straightforward:

• First, apply the square root property by taking the square root of both sides of the equation.
• Once the square root is taken, don't forget that there are typically two solutions due to the plus-minus sign (\(\pm\)).
• Isolate the variable \(x\) by moving constants from one side to the other. If you end with equations like \(x+1 = \sqrt{5}\), solve them by subtracting 1 from each side, yielding \(x = \sqrt{5} - 1\) and \(x = -\sqrt{5} - 1\).
• The solutions obtained provide the values of \(x\) that satisfy the original quadratic equation.

Using such systematic approaches helps in solving even the most challenging quadratic equations efficiently, giving you a reliable method to find solutions.