Problem 45 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+4 x+1=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(x^{2}+4 x+1=0\) by completing the square are \(x=-2+\sqrt{3}\) and \(x=-2-\sqrt{3}\).

Step 1: Rearranging the Equation

First, isolate \(x\)-terms on one side by subtracting 1 from both sides, resulting in the equation \(x^{2}+4 x=-1\).

Step 2: Find the Value that Completes the Square

To convert the quadratic expression to a perfect square trinomial, we need to add a certain value to it. This can be done by taking half the coefficient of \(x\), squaring it and adding to both sides. Here, the coefficient of \(x\) is 4, and its half is 2. Squaring 2, we get 4. Now, add this to both sides to get \(x^{2}+4x+4=4-1\).

Step 3: Simplify the Equation

Simplify the expressions on both sides to get the equation in the form of a perfect square trinomial. The revised equation becomes \((x+2)^2=3\).

Step 4: Solving for \(x\)

Take square root on both sides to solve for \(x\). It results in two solutions, \(x=-2+\sqrt{3}\) and \(x=-2-\sqrt{3}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomials of degree two and generally have the standard form \( ax^2 + bx + c = 0 \). Solving these equations is a fundamental aspect of algebra. One effective method is 'completing the square'. This involves transforming the equation into a perfect square trinomial to make it easier to solve. Here's a distilled overview on how to do it:

• Rearrange the quadratic equation so that the terms with the variable are on one side and the constant is on the other side.
• Find the number that completes the square for the binomial and add it to both sides, balancing the equation.
• Rewrite the resulting side with the variable as the square of a binomial.
• Solve for the variable by taking the square root of both sides and then isolating the variable.

In the exercise provided, \( x^{2}+4 x+1=0 \), the method reveals two solutions: \( x=-2+\sqrt{3} \) and \( x=-2-\sqrt{3} \) after applying the completing the square technique. This brings us to the realization that quadratic equations might yield two real answers due to the square root step involved.

Perfect Square Trinomials

Understanding perfect square trinomials is vital in simplifying the process of solving quadratic equations. A perfect square trinomial is formed when a binomial is squared, following the pattern \( (ax + b)^2 = a^2x^2 + 2abx + b^2 \). To identify or create one:

• Look for a trinomial where the first and last terms are squares themselves.
• Ensure the middle term is twice the product of the square roots of the first and last terms.
• If the trinomial doesn't meet the criteria, calculate and add the necessary constant to make it a perfect square on both sides of the equation (as seen in the completing the square method).

In the given exercise, forming a perfect square trinomial led to \( (x+2)^2 \) which is a pivotal step as it allowed us to segue into easier ways of reaching the solution, by later applying the square root to find the values of \( x \).

Algebraic Techniques

Algebraic techniques such as completing the square are handy tools in solving not just quadratic equations but in many other mathematical contexts. It is important to understand various algebraic techniques and know when to employ them. Other techniques include:

• Factoring
• Using the quadratic formula
• Graphing

Each technique offers its own advantages. While factoring might be the simplest approach for some equations, it might not always be possible. Graphing provides a visual solution, but might not be accurate for precise answers. The quadratic formula is a reliable catch-all method but can be more complex. Completing the square is especially useful when dealing with equations that lead to a perfect square trinomial, as it encourages a logical step-by-step progression towards the solution. Its mastery can significantly simplify the process and enhance a student's algebraic problem-solving abilities.