Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. $$ 4 x^{2}-15=25 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = \sqrt{10}\) and \(x = -\sqrt{10}\).

Step 1: Rewrite Equation

Begin by isolating the quadratic part of the equation. This involves moving the constants to the other side of the equation. Hence, the equation can be rewritten as: \(4x^2 = 25 + 15\) which simplifies to \(4x^2 = 40\).

Step 2: Simplify Equation

To ease the process of further calculation, divide the whole equation by the coefficient of \(x^2\), which is 4. This yields: \(x^2 = 10\).

Step 3: Apply Square Root Property

With the equation \(x^2 = 10\), applying the square root property leads to two solutions. The square root property states that if \(p^2 = q\), then \(p = \sqrt{q}\) or \(p = -\sqrt{q}\). In the context of the equation, \(x = \pm \sqrt{10}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are those in which the highest power of the unknown variable is 2. They're often written in the form \( ax^2 + bx + c = 0 \). Our task is to find the value of the variable that makes this expression true. In simple terms, solving quadratic equations means finding the value(s) of \( x \) that satisfy the equation. Methods for solving include:

• Factoring the equation
• Using the square root property
• Completing the square
• Applying the quadratic formula

Each method has its own approach and is useful depending on the form of the equation. For example, equations that are easy to rearrange into squares are ideal for the square root property method. Whenever you approach a quadratic equation, it's helpful to quickly assess the simplest method to apply based on its current form.

Square Root Property

The square root property is a specific technique used to solve equations that have been reduced to the form \( x^2 = k \). It’s particularly useful when other methods, like factoring, are not straightforward. This property states that if \( p^2 = q \), then \( p \) equals the positive or negative square root of \( q \), or \( p = \pm \sqrt{q} \). When applying this method:

• Solve for \( x^2 \) so the equation looks like \( x^2 = k \).
• Apply the square root to both sides, remembering that square roots have two potential results: positive and negative.
• Simplify to find \( x \).

This results in two possible solutions, representing both the positive and the negative roots. It’s crucial because every squared number has both a positive and a negative root, and considering both ensures that all potential solutions are accounted for.

Equation Simplification

Equation simplification is the process of manipulating an equation to make it easier to solve. This often involves basic arithmetic operations and rearranging terms. Key steps to simplify a quadratic equation can include:

• Moving constant terms from one side of the equation to the other using addition or subtraction, ensuring the quadratic term stands alone as much as possible.
• Dividing the whole equation by a common factor, particularly useful if the coefficients are large or if you're trying to isolate \( x^2 \).
• Reducing terms and fractions to their simplest form.

In the original problem, simplifying the equation \( 4x^2 - 15 = 25 \) involved first moving the constant 15 to the other side and then dividing by 4, leading us to \( x^2 = 10 \). This makes it much easier to either apply the square root property or another method of solving. The goal is always to make \( x \) as accessible as possible for solving.