Problem 44 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-2 x-5=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-2x-5=0\) by completing the square are \(x=1+\sqrt{6}\) and \(x=1-\sqrt{6}\)

Step 1: Rearrange the Equation

Rewrite the equation so that only the terms involving \(x\) are on one side. The equation becomes: \(x^{2}-2x=5\)

Step 2: Complete The Square

The formula of a perfect square is \(a^2-2ab+b^2\). In this case, \(a\) is \(x\), and \(b\) should be chosen such that it fulfills the formula. This means \(b^2\) should be added to both sides of the equation. To find \(b\), divide the coefficient of the \(x\) term by \(2\), which is \(-2/2=-1\). Hence \(b\) is \(-1\) and we add \(b^2=(-1)^2=1\) to both sides of the equation: \(x^{2}-2x+1=5+1\)

Step 3: Rewrite as a Binomial Squared

The left side of the equation is now a perfect square and can be written as a binomial squared. This gives \( (x-1)^2=6 \)

Step 4: Solve for \(x\)

Expand the square root of both sides: \( x-1=\pm\sqrt{6} \). Hence, there are two possible solutions for \( x: x=1+\sqrt{6} \) and \( x=1-\sqrt{6} \)

Key Concepts

Algebraic Equations

Algebraic equations form the foundation of many mathematical concepts and are essential tools for solving a variety of problems. These equations are formed by equating two algebraic expressions and often involve variables which represent unknown values that we need to solve for. An example of a simple algebraic equation would be something like, \( 2x + 3 = 7 \), where we would solve for the value of \( x \) to make the equation true.

To solve algebraic equations, we apply arithmetic operations and algebraic principles like combining like terms and using the distributive property to isolate the variable. In more advanced scenarios, we use techniques such as completing the square, particularly for solving quadratic equations, to transform equations into a more manageable form that makes finding solutions straightforward. The importance of understanding algebraic equations cannot be overstated as they are not just a fundamental part of algebra, but of mathematics as a whole.

Quadratic Equations

Understanding Quadratic Equations

Quadratic equations are a special type of algebraic equations where the degree, or the highest power of the variable, is two. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \) is not equal to zero. These equations describe parabolas when graphed and can intersect the x-axis at zero, one, or two points corresponding to the number of real solutions the equation has.

Solving Quadratic Equations

Several methods exist for solving quadratic equations, including factoring, using the quadratic formula, graphing, and completing the square. The method of completing the square is particularly insightful, as it can always be used to find a solution and serves as the derivation for the quadratic formula itself. This method transforms the original equation into the form of a perfect square trinomial, which then can be easily solved by taking the square root of both sides of the equation.

Perfect Square Trinomials

Perfect square trinomials are polynomial expressions that are the squares of binomials. In other words, when a binomial is multiplied by itself, it results in a perfect square trinomial. The generic form is \( (a \pm b)^2 = a^2 \pm 2ab + b^2 \). Recognizing and creating these trinomials is key to solving quadratic equations by the method of completing the square.

To complete the square, one must find a value \( b \) that when added and subtracted to the existing equation forms such a trinomial. This involves halving the coefficient of the middle term and squaring the result, then adding and subtracting this square to/from the equation in an balanced way to maintain the equation's equality. Afterward, the equation can be written as the square of a binomial, greatly simplifying the solution process. This technique not only aids in solving equations, but also enhances understanding of the structure of algebraic expressions and their geometric representations.