Problem 15 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}-10 x=-24\)

Step-by-Step Solution

Answer

The solutions for the equation \(x^{2}-10 x = -24\) are \(x = -6\) and \(x = -4\).

Step 1: Rearrange the Equation

First, the quadratic equation needs to be written in a standard form. To this end, rewrite the equation so that the right-hand side equals zero: \(x^{2}-10 x + 24 = 0\).

Step 2: Identify a, b, and c

In the standard form of a quadratic equation, \(ax^2+bx+c=0\), a is the coefficient of \(x^2\), b is the coefficient of x and c is the constant. Here, a = 1, b = -10, and c = 24.

Step 3: complete the square

To complete the square, we rewrite the equation as \((x+\frac{b}{2a})^2 - (\frac{b}{2a})^2 + c = 0\). Plugging our values for a, b, and c, we get \((x+5)^2 - 25 + 24 = 0\). Simplify this to \((x+5)^2 = 1\).

Step 4: Solve for x

Now that the equation is in \((x+\frac{b}{2a})^{2}=c\) form, solve for x. This involves taking the square root of both sides, to find \(x = -5 +- 1\). Therefore, the solutions for x are -6 and -4.

Key Concepts

Quadratic Equations

Quadratic equations are an essential component in algebra, characterized by an equation of degree 2, with the highest exponent of the variable being 2. The general form of a quadratic equation is written as \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The values of \(a\), \(b\), and \(c\) can affect the shape and position of the parabola that the equation represents when plotted on a graph.

These equations often appear in various applications, such as physics, engineering, and economics, due to their ability to model parabolic paths, like the trajectory of a projectile. Understanding how to solve these equations is crucial for students not only to succeed in math courses but also to apply mathematical reasoning to real-world problems.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is pivotal for understanding how to solve for \(x\). It is expressed as \(ax^{2} + bx + c = 0\), where each term is on one side of the equation, and the other side is set to zero.

Properties

The coefficient \(a\) influences the width and direction of the parabola—whether it opens upwards or downwards. The \(b\) coefficient affects the position of the parabola along the x-axis, while \(c\) represents the y-intercept. Reorganizing an equation into this form sets the stage for solving the equation using several methods, including completing the square, which centers on the concept of creating a perfect square trinomial on one side of the equation.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. There are multiple techniques available to tackle these equations, each with its benefits and best-use scenarios.

Methods of Solving Quadratics

• The Quadratic Formula: An all-encompassing method that gives the roots directly.
• Factoring: This method is efficient when the quadratic can be easily decomposed into the product of binomials.
• Graphing: Useful for visualizing the solutions as the x-intercepts of the parabola.
• Completing the Square: This method transforms the equation into a perfect square, making it straightforward to solve for \(x\).

Choosing the appropriate approach depends on the equation's complexity and the specific context in which you are solving it.

Completing the square, the focus of our exercise, offers a clear path to the solution when other methods are not applicable or when students need to derive the quadratic formula itself. By mastering multiple solving methods, students become more versatile in handling various types of quadratic equations.

Algebraic Methods

Algebraic methods refer to the techniques used to manipulate and solve algebraic equations, including quadratic equations. A fundamental aspect of algebraic methods is the use of symbolic representations to abstractly denote problems and solutions.

Key Principles

• Combining Like Terms: Simplify expressions by adding or subtracting similar terms.
• The Distributive Property: Allows for the multiplication across parentheses.
• Transposition: The movement of terms from one side of an equation to another.
• Factoring: Breaking down complex expressions into simpler components.

A systematic approach using these principles often leads to the solution of an equation. In the context of our quadratic equation exercise, completing the square is an algebraic method that carefully arranges terms to facilitate the extraction of roots. Understanding these principles fuels a student's ability to tackle increasingly complex algebraic challenges effectively.