Problem 53 - Algebra and Trigonometry

Question

Solve equation by completing the square. $$ x^{2}+4 x+1=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(x^{2}+4 x+1=0\) are \(x1 = \sqrt{3} - 2\) and \(x2 = -\sqrt{3} - 2\).

Step 1: Rearrange the Equation

First, the equation needs to be rearranged as \(x^2 + 4x = -1\) to create room for completing the square.

Step 2: Complete the Square

The goal in completing the square is to transform the equation into a perfect square trinomial on one side. To do this, add \((4/2)^2 = 4\) to both sides of the equation. This yields \(x^2 + 4x + 4 = 4 - 1 = 3\). The left part of the equation is a perfect square trinomial which can be expressed as \((x+2)^2\). Hence, the equation becomes \((x+2)^2 = 3\).

Step 3: Apply the Square Root Rule

Apply the square root rule to solve for \(x\). This rule is expressed as \(\sqrt{(x+2)^2}=\sqrt{3}\). This gives \(x + 2 = \sqrt{3}\) and \(x + 2 = -\sqrt{3}\), hence \(x1 = \sqrt{3} - 2\) and \(x2 = -\sqrt{3} - 2\).

Key Concepts

Quadratic Equations

Quadratic equations are foundational in algebra and they take the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The solutions to these equations, also known as 'roots', are the values of \(x\) that make the equation true. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, graphing, and completing the square.

Completing the square is particularly useful when the quadratic doesn't factor neatly or when you aim to derive the quadratic formula itself. This method transforms the original quadratic equation into the form of \((x+p)^2=q\), which can then be solved by taking the square root of both sides. The results can be simplified to obtain the roots of the original equation.

Perfect Square Trinomial

A perfect square trinomial is formed when a binomial is squared, resulting in the squared terms for the first and last term, and a middle term that equals twice the product of the two terms of the binomial. A perfect square trinomial can always be expressed as \((x + n)^2\) or \((x - n)^2\), where \(n\) is a real number.

In the context of completing the square for solving quadratic equations, finding the correct value to add to the equation to form a perfect square trinomial is key. If you have an equation like \(x^2 + bx\), the value to add to both sides is \((\frac{b}{2})^2\), effectively completing the square. Recognizing these trinomials is a valuable skill in algebra, as it makes it easier to solve quadratic equations.

Square Root Rule

The square root rule is an algebraic technique used to solve equations where the variable is squared. It is applied after creating a perfect square trinomial by completing the square in a quadratic equation. According to the square root rule, if you have an equation in the form \((x + n)^2 = m\), you can take the square root of both sides to solve for \(x\), resulting in \(x + n = \sqrt{m}\) and \(x + n = -\sqrt{m}\) because both the positive and negative square roots of a number are valid.

This method provides a straightforward pathway to finding the roots of the equation without resorting to the quadratic formula. However, it is crucial to include both the positive and negative solutions to accurately reflect all solutions for \(x\).

Algebraic Techniques

Fundamentals of Algebraic Manipulation

Algebraic techniques encompass various tools that allow us to manipulate equations and expressions to solve for unknown variables. These include operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression, which maintains the equality.

Completing the Square in Practice

When it comes to completing the square, algebraic skills are essential. The technique requires the ability to rearrange terms, carefully balance equations by performing the same operation on both sides, and identifying when a trinomial can become a perfect square. Mastery of these algebraic techniques can simplify complex problems and is instrumental in higher-level mathematics, such as calculus and beyond. Being adept in algebra opens up a wide range of problem-solving strategies, making it a critical component of a student's mathematical toolkit.