Problem 17 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}-2 x=5\)

Step-by-Step Solution

Answer

Hence, the solutions of the given quadratic equation are \(x = 1 + \sqrt{6}\) and \(x = 1 - \sqrt{6}\)

Step 1: Rearrange the Equation

Start by moving the constant term to the other side of the equation to make room for completing the square. So, \(x^{2}-2 x = 5\) will be rearranged as \(x^{2}-2 x - 5 = 0\)

Step 2: Complete the Square

The next step is to complete the square. This involves adding the square of half the coefficient of \(x\) to both sides of the equation. Since the coefficient of \(x\) is -2, half of it will be -1 and its square will be 1. So we add 1 to both sides to get \(x^{2}-2 x + 1 = 5 + 1\) which simplifies to \((x-1)^{2} = 6\)

Step 3: Find the Value of x

Now, you have an equation in a form that makes it easy to solve for \(x\). To obtain \(x\), take a square root of both sides to get \(x-1 = \sqrt{6}\) or \(x-1 = -\sqrt{6}\). Solving both equations gives \(x = 1 + \sqrt{6}\) and \(x = 1 - \sqrt{6}\) respectively

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form of \( ax^2 + bx + c = 0 \). This means it has the variable raised to the power of two, which is why it's called "quadratic." The equation could look something like \( x^2 - 2x = 5 \) as in the original exercise. In this form, quadratic equations can model various scenarios in math and the real world, such as projectile motion or area problems.

Quadratic equations can be solved using several methods including factorization, the quadratic formula, and completing the square. Each method has its advantages in different situations. Recognizing the structure of the equation helps in deciding which technique will be most efficient for solving it, especially when equations are not easily factored. This flexibility makes quadratic equations an essential part of algebraic education.

Learning to solve quadratic equations helps in developing critical problem-solving skills that are applicable in various fields like physics, engineering, and finance.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. For quadratic equations like \( x^2 - 2x - 5 = 0 \), solving means determining the value(s) of \( x \) that satisfy the equation. One way to solve this is by rearranging the equation and isolating \( x \). This process often starts by moving terms around to make algebraic manipulation easier.

When completing the square, the goal is to rewrite the quadratic equation into a perfect square trinomial form, like \( (x - 1)^2 = 6 \). This is done by adding or subtracting necessary terms to both sides of the equation. Solving the resulting perfect square trinomial allows you to use simple algebraic operations, like taking the square root, to find the values of \( x \).

• Rearranging the equation makes it easier to identify the terms needed for completing the square.
• Taking square roots is a straightforward step once a perfect square is achieved.

These steps highlight the importance of understanding the underlying structure of equations in order to solve them effectively.

Algebraic Techniques

Algebraic techniques are methods used to manipulate and solve equations. Completing the square is a specific algebraic technique that converts a quadratic equation into a form that is easier to solve. It relies on adding and subtracting the correct constant to transform part of the equation into a perfect square trinomial.

Here's how it works: given an equation like \( x^2 - 2x = 5 \), you first rewrite it by rearranging terms if necessary. Then, you complete the square by adding \( 1 \) (since it is \( (-1)^2 \)) to both sides, resulting in \( x^2 - 2x + 1 = 6 \). This converts the equation into \( (x-1)^2 = 6 \), which is a more straightforward form.

Using techniques such as completing the square:

• Provides a systematic approach to rewriting equations.
• Makes it easier to isolate the variable of interest.

Mastery of algebraic techniques not only helps in solving equations but also builds a strong foundation for understanding more complex algebraic concepts.