Problem 80 - Elementary Algebra

Question

Set up an algebraic equation and use it to solve the following. If 20 is subtracted from the square of a number, then the result is \(4 .\) Find the number.

Step-by-Step Solution

Answer

The number is \( \pm 2\sqrt{6} \).

Step 1: Define the Variable

Let's represent the number we are trying to find with the variable \( x \). Our goal is to find the value of \( x \).

Step 2: Form the Equation

According to the problem, if 20 is subtracted from the square of a number, the result is 4. Mathematically, we can express this as: \( x^2 - 20 = 4 \).

Step 3: Isolate the Square Term

Add 20 to both sides of the equation to isolate the square term. This gives us \( x^2 = 4 + 20 \), which simplifies to \( x^2 = 24 \).

Step 4: Solve for the Variable

To solve for \( x \), take the square root of both sides of the equation. We have \( x = \pm \sqrt{24} \). Simplifying, we get \( x = \pm 2\sqrt{6} \).

Step 5: Check the Solutions

Verify by substituting \( x = 2\sqrt{6} \) and \( x = -2\sqrt{6} \) back into the original condition. Both values satisfy the equation since subtracting 20 from their squares results in 4.

Key Concepts

Variable Definition

The first step to solving algebraic equations is defining what we're trying to find. In problems like the one presented, where you need to find an unknown number, it's common to use a variable to represent it. A variable is typically a letter, often \( x \), that stands in place of the unknown number. Here, we define \( x \) as the number we are searching for. The variable definition is crucial because it transforms a word problem into an algebraic equation that can be solved systematically. By knowing what \( x \) represents, we can set up an equation that models the situation described in the problem.

Isolation of Terms

Once the equation is set up, we need to manipulate it to make it easier to solve. This often involves **isolating terms**. In our example, the equation is \( x^2 - 20 = 4 \). We aim to isolate \( x^2 \), the square term. To do this, perform arithmetic operations to 'move' other terms to the opposite side of the equation. Here, by adding 20 to both sides, the equation simplifies to \( x^2 = 24 \). Isolation is a powerful technique because it helps reduce complexity and focuses on what needs to be solved. It is precisely these systematic steps that lead us closer to the solution.

Square Roots

After isolating the square term, the next step is to find the value of \( x \). Since \( x^2 = 24 \), we take the square root of both sides to solve for \( x \). This gives us \( x = \pm \sqrt{24} \).

It's important to note that squaring can result in a positive or negative number, hence the \( \pm \) sign. Simplifying \( \sqrt{24} \), we factor it to \( \sqrt{4 \times 6} = 2\sqrt{6} \), which gives us \( x = \pm 2\sqrt{6} \). Understanding square roots is vital because it allows us to solve equations where the variable is squared, opening up a wider range of solutions.

Quadratic Equations

Quadratic equations are those where the variable is raised to the power of two (squared), as seen in \( x^2 = 24 \). These equations can have up to two solutions due to the nature of squaring. In our case, both \( 2\sqrt{6} \) and \( -2\sqrt{6} \) make the original equation true when substituted back.

Quadratics can present various forms of equations but share a central feature of having a squared term. Solving quadratics often involves techniques like factoring, completing the square, or using the quadratic formula. Recognizing and solving quadratic equations is a key skill in algebra, providing solutions to numerous real-world problems.