Problem 38 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \((x-5)(x-3)=1\)

Step-by-Step Solution

Answer

The roots of the quadratic equation are \(x = 4 + \sqrt{2}\) and \(x = 4 - \sqrt{2}\).

Step 1: Expand

First, the equation \((x-5)(x-3)=1\) should be expanded to get a quadratic equation in the form of \(ax^2 + bx + c\). Expand the parentheses, which leaves \(x^2 - 8x + 15 - 1 = 0\). So the equation turns into \(x^2 - 8x + 14 = 0\).

Step 2: Subtract 14 from both sides

The next step is to subtract 14 from both sides, which gives an equation \(x^2 - 8x = -14\). This is done to prepare to complete the square.

Step 3: Adding Square term

The coefficient of \(x\) is divided by 2 and squared. The result is \(16\). This will be added to both sides of the equation. This gives \(x^2 - 8x + 16 = -14 + 16\).

Step 4: Rewrite and simplify equation

The equation can now be rewritten in simplified form. On the left side, a square of binomial form will appear. This gives \((x - 4)^2 = 2\).

Step 5: Solve for x

Finally, the equation can be solved for \(x\). This requires taking the square root of both sides, keeping both the positive and negative solutions. This results in \(x - 4 = \sqrt{2}\) and \(x - 4 = -\sqrt{2}\). Solve both equations separately for \(x\), which results in \(x = 4 + \sqrt{2}\) and \(x = 4 - \sqrt{2}\).

Key Concepts

Quadratic Equation

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. These equations are called quadratic because "quad" refers to a square, as the highest degree of the variable \(x\) is 2.

Quadratic equations can be solved using various methods, such as factoring, using the quadratic formula, or completing the square. Each method has its own advantages depending on the specific equation. In this exercise, we focus on the method known as "completing the square." Understanding this method provides a strong foundation for solving more complex quadratic equations and helps grasp the origins of other methods like the quadratic formula.

Recognizing a quadratic equation is simple: look for the highest power of 2 on the variable. Expanding expressions or simplifying terms will help to get the equation into standard quadratic form, which is essential for solving it. In our example, \((x-5)(x-3)=1\) was expanded to form \(x^2 - 8x + 14 = 0\). This gives us a starting point to apply the completing the square method.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of the variable that makes the equation true. With quadratic equations, specific techniques are used to isolate the variable in a way that allows us to find its value accurately.

A structured approach is crucial, especially when dealing with equations like \(x^2 - 8x = -14\). Begin by manipulating the equation to prepare it for completing the square. This involves isolating terms that include the square variable and constant terms on opposite sides.

Solving an equation by completing the square requires a few strategic steps:

• Rewrite the equation so the square term and linear term are on one side.
• Add a number to both sides to transform the quadratic term into a perfect square trinomial.
• Decompose the perfect square trinomial into a binomial squared.
• Use inverse operations, such as taking the square root, to solve for the variable.

Applying these steps to \(x^2 - 8x = -14\), we eventually obtain \((x - 4)^2 = 2\). Solving this gives two values for \(x\), reflecting the parabolic nature of quadratic equations.

Algebra Steps

Algebraic steps taken to solve equations effectively depend on understanding how to manipulate expressions logically and systematically. When completing the square, these steps become clearer with practice.

First, after expanding or simplifying to achieve the standard quadratic form, the next critical step is preparing the equation for the square. We subtract any constant sitting on the right to make the equation ready for completing the square. In the exercise example, this was achieved when setting up \(x^2 - 8x = -14\).

To complete the square:

• Take half the coefficient of \(x\), square it, and add it to both sides of the equation. In our example, \(-8/2 = -4\), and \((-4)^2 = 16\) was added to both sides.
• This turns the left-hand side into a perfect square trinomial, which can be factored as \((x - 4)^2\).
• Finally, solve the resulting simple equation by taking the square root of both sides. Always remember to consider both the positive and negative roots, yielding \(x = 4 + \sqrt{2}\) or \(x = 4 - \sqrt{2}\).

By breaking down these algebra steps one by one, we ensure the solution is accurate and easier to understand. Practice makes these processes intuitive, aiding in solving more complex algebraic problems with ease.