Problem 15 - Algebra and Trigonometry

Question

Solve the equation by factoring. $$ x^{2}=5(x+100) $$

Step-by-Step Solution

Answer

The solutions are \(x = 25\) and \(x = -20\).

Step 1: Expand the Right Side

Start by expanding the right side of the equation. The given equation is \(x^2 = 5(x + 100)\). Distribute the 5 across the terms inside the parentheses:\[x^2 = 5x + 500\]

Step 2: Move All Terms to One Side of the Equation

To solve by factoring, we need all terms on one side of the equation. Subtract \(5x + 500\) from both sides to set the equation to zero:\[x^2 - 5x - 500 = 0\]

Step 3: Factor the Quadratic Equation

Now, factor the quadratic equation \(x^2 - 5x - 500\). We are looking for two numbers that multiply to -500 and add to -5. The numbers that satisfy this condition are 20 and -25. So, the factorization is:\[(x - 25)(x + 20) = 0\]

Step 4: Solve for the Roots

Set each factor equal to zero and solve for \(x\).1. \(x - 25 = 0\) implies \(x = 25\).2. \(x + 20 = 0\) implies \(x = -20\).

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable. It typically takes the form \( ax^2 + bx + c = 0 \). In our exercise, the quadratic equation is \( x^2 - 5x - 500 = 0 \). The term quadratic comes from "quad" meaning square, indicating the \( x^2 \) term is the highest degree. Quadratic equations often describe parabolas when graphed on a coordinate plane.

In this equation:

• \( x^2 \) is the quadratic term.
• \(-5x \) is the linear term.
• \(-500 \) is the constant term.

These equations can often be solved by factoring, completing the square, or using the quadratic formula. Here, we have solved it by factoring, focusing on rearranging the original equation to help us identify the solution as it transforms into a product of binomials.

Solving Equations

Solving equations means finding the value(s) of the unknown variable(s) that will satisfy the equation. For quadratic equations, this often includes methods like factoring, using the quadratic formula, or completing the square.

In the current exercise, we used factoring to solve the equation, which involves several key steps:

• Expanding or simplifying the equation to get all terms on one side, \( x^2 - 5x - 500 = 0 \).
• Factoring the quadratic expression to find two binomials (\( x - 25 \) and \( x + 20 \)).
• Setting each binomial equal to zero to solve for the variable \( x \).

The goal is to express the quadratic as a product of two simpler expressions set to zero. Once in this form, each factor can reveal potential solutions, often referred to as "roots." These steps collectively transform the initial problem into workable solutions that define where the expression evaluates to zero.

Roots of Equations

The roots of an equation are the values of the variable that make the equation true, often meaning they reduce it to zero. In a quadratic equation like \( ax^2 + bx + c = 0 \), there can be two solutions, one solution, or no real solutions, depending on the discriminant \( b^2 - 4ac \).

The exercise reveals the roots of the equation \( x^2 - 5x - 500 = 0 \) are:

• \( x = 25 \)
• \( x = -20 \)

These roots were found after setting each factor from the factored form, \((x - 25)(x + 20) = 0\), equal to zero. This process highlights the importance of factoring, as it simplifies the equation into components that directly equate to the solutions.

Interestingly, these roots represent the points where the parabola defined by the quadratic will intersect the x-axis when graphed. This visual interpretation can often aid in understanding how changes to coefficients affect the equation's graph and its solutions.