Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ x^{2}+4 x=21 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = -7\) and \(x = 3\).

Step 1: Rewrite the Quadratic Equation

Rearrange the equation \( x^2 + 4x = 21 \) to get it in standard form \( ax^2 + bx - c = 0 \). This results in \( x^2 + 4x - 21 = 0 \).

Step 2: Factor the Quadratic Equation

Factor the equation. Look for two numbers that multiply to \(-21\) and add to \(4\). The numbers \(7\) and \(-3\) fit these conditions. Write the factors as \( (x + 7)(x - 3) = 0 \).

Step 3: Solve for x

Set each factor equal to zero and solve for \(x\). The solutions are \( x = -7 \) and \( x = 3 \).

Key Concepts

Factoring

Factoring is a technique used in algebra to simplify expressions and solve equations. When it comes to quadratic equations, factoring is often employed to find the roots or solutions of the equation. In the context of quadratic equations, factoring involves expressing the quadratic expression as a product of linear expressions. For example, the quadratic expression \( x^2 + 4x - 21 \) can be factored into \((x + 7)(x - 3)\).

The goal of factoring is to break down the quadratic into simpler pieces that multiply to form the original expression. This process requires finding two numbers that multiply to the product of the coefficient of the squared term and the constant term (in this case \(-21\)) and also add up to the coefficient of the linear term (which is \(4\)).

* **Benefits of Factoring:** * Simplifies the problem
* Makes solving for variable values straightforward

Once the expression is factored, the equation becomes easier to solve, as you simply set each linear factor equal to zero and solve for the variable.

Standard Form

The standard form of a quadratic equation is crucial for solving and manipulating these equations. It’s written as \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Getting the equation into standard form is the first step in many solving techniques, such as factoring or using the quadratic formula.

To convert the given equation \( x^2 + 4x = 21 \) into standard form, you need to move all terms to one side of the equation, resulting in \( x^2 + 4x - 21 = 0 \). This rearrangement sets the equation up for further solving methods and highlights the coefficients that will be crucial in factoring or using the quadratic formula.

Using standard form simplifies the equation and allows for easier identification of the quadratic's properties, such as its direction of opening, vertex, and intercepts.

Solving Quadratics

There are various methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. In our exercise, the easiest method is factoring, especially when the equation is easy to decompose into linear factors.

Once a quadratic equation like \( x^2 + 4x - 21 = 0 \) is factored into \( (x + 7)(x - 3) = 0 \), solving it becomes straightforward. You take each factor and set them equal to zero:

• \( x + 7 = 0 \)
• \( x - 3 = 0 \)

By solving these simple equations, you find that \( x = -7 \) and \( x = 3 \). These solutions are the x-intercepts of the related quadratic function when graphed, showing where it crosses the x-axis.

* **Methods for Solving Quadratics:** * Factoring
* Quadratic formula
* Completing the square

Each method has its advantages, depending on the specific form and coefficients of the quadratic equation.