Problem 56 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(5 x^{2}-125=0\)

Step-by-Step Solution

Answer

The real solutions are \(x = 5\) and \(x = -5\).

Step 1: Rewrite the Equation

Start with the given equation: \[ 5x^2 - 125 = 0 \]Add 125 to both sides to isolate the term with the variable, resulting in:\[ 5x^2 = 125 \]

Step 2: Simplify the Equation

Divide both sides of the equation by 5 to solve for \(x^2\):\[ x^2 = \frac{125}{5} = 25 \]

Step 3: Solve for x

To find the value of \(x\), take the square root of both sides:\[ x = \pm \sqrt{25} \]Thus:\[ x = \pm 5 \]

Step 4: Write the Real Solutions

The real solutions for the equation are found to be:\[ x = 5 \] and \[ x = -5 \]

Key Concepts

Solving Quadratic Equations

Solving a quadratic equation involves a few straightforward yet essential steps. Quadratic equations are of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our exercise, the quadratic equation is given as \(5x^2 - 125 = 0\). Our goal is to find the values of \(x\) — the unknowns.
To start off, the equation needs to be simplified by isolating the term containing the variable. This means we want to get \(x^2\) by itself on one side of the equation.

• First, we add or subtract terms to move the constant part of the equation to the opposite side.
• Next, we eliminate any coefficients attached to \(x^2\) by dividing the entire equation. This leaves us with \(x^2\) on one side and a number on the other side.

Once this is accomplished, the equation is ready for solving the unknown \(x\). We can now proceed to evaluate \(x\) using other methods, like the square root, as we'll see in the following sections.

Understanding the Square Root

The square root is a crucial concept for solving equations involving squares, such as our equation \(x^2 = 25\). Taking the square root is the process of determining which number multiplied by itself gives us the original number. In essence, we are reversing the process of squaring a number.
When the equation \(x^2 = 25\) is established, we apply the square root operation to both sides to simplify it further.

• This involves finding numbers that when squared result in 25. Specifically, we look for numbers \(x\) such that \(x \times x = 25\).
• Remember that both a positive and a negative number can fulfill this condition, because \((-5) \times (-5) = 25\) as well.
• Therefore, we represent the solution as \(x = \pm\sqrt{25}\), deducing that \(x = 5\) or \(x = -5\).

Understanding the concept of taking square roots helps unlock the answers to many quadratic equations effectively and efficiently.

Identifying Real Solutions

Identifying real solutions is a pivotal step in solving quadratic equations. Real solutions refer to those solutions that are real numbers rather than imaginary or complex numbers.
In our equation \(5x^2 - 125 = 0\), we've established real solutions after taking the square root of both sides. The terms \(x = 5\) and \(x = -5\) are real numbers, fitting comfortably within the realm of real solutions.

• Real numbers include all the rational and irrational numbers that we are used to working with in everyday mathematics.
• Real solutions are especially crucial in situations modeled by physics or everyday problems where we need answers that make sense practically.
• By verifying each step in the quadratic equation, we can be confident that the values we find not only satisfy the equation but are tangible and applicable numbers.

When we solve quadratic equations like this, identifying the real solutions ensures that we find meaningful answers that can be utilized in further analysis or application.