Problem 64 - Elementary Algebra

Question

Find a quadratic equation in standard form with the following solutions. $$ \pm 13 $$

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 - 169 = 0\).

Step 1: Identify the Solutions

The given solutions are \( x = 13 \) and \( x = -13 \). These are the roots of the quadratic equation we need to find.

Step 2: Use Factored Form of Quadratic Equation

Since \( x = 13 \) and \( x = -13 \) are the solutions, we can express the quadratic equation in factored form as \((x - 13)(x + 13) = 0\).

Step 3: Expand the Factored Form

Expand the expression \((x - 13)(x + 13)\) using the identity \((a-b)(a+b) = a^2-b^2\). This gives us: \(x^2 - 13^2 = x^2 - 169\).

Step 4: Write in Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). So, the equation \(x^2 - 169 = 0\) is already in the standard form where \(a = 1\), \(b = 0\), and \(c = -169\).

Key Concepts

Factored Form

Quadratic equations are often easier to work with when expressed in their factored form. The factored form of a quadratic equation is represented as

• (a)(b) = 0This equation suggests that either of the factors, (x - r) or (x - s), can be zero for the equation to hold true.For our problem, the solutions are given as 13 and -13. This means that the quadratic equation can be written as \((x - 13)(x + 13) = 0\).The factored form is particularly useful because it directly provides the roots of the equation, revealing where the parabola intersects the x-axis.This form is also beneficial for solving the equation, checking the work, or converting to other forms, such as the standard form by expanding the terms.

Standard Form

The standard form of a quadratic equation is fundamental in understanding the equation's structure. It is represented as \( ax^2 + bx + c = 0 \).Here, coefficients a, b, and c are constants. They represent the squared term, the linear term, and the constant part of the equation, respectively.In our exercise, expanding the factored form \((x - 13)(x + 13)\) using the identity \((a - b)(a + b) = a^2 - b^2\) results in:\[x^2 - 169 = 0,\]which is already in standard form, where:

• a = 1
• b = 0
• c = -169

Being in standard form helps in identifying these coefficients, assisting in further analysis or transformations, such as finding the vertex or converting it back into factored form.

Roots of Equations

The roots of a quadratic equation are the values of x for which the equation equals zero, essentially where the parabola crosses the x-axis. These roots are also referred to as solutions or zeros. Finding the roots is often a key part of solving quadratic problems.In this exercise, the given roots are \( x = 13 \) and \( x = -13 \). Knowing the roots upfront allows the quadratic equation to be accurately constructed in its factored form: \((x - 13)(x + 13) = 0\).Understanding how to derive a quadratic equation from its roots can aid in more complex problems, such as those involving higher degrees, different variable relationships, or applications in physics and engineering, where these principles are frequently used.