Problem 49 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{2}-24=0 $$

Step-by-Step Solution

Answer

The real solutions are \(x = 2\sqrt{6}\) and \(x = -2\sqrt{6}\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \(x^2 - 24 = 0\). Quadratic equations are polynomials of degree 2.

Step 2: Isolate the Quadratic Term

To solve \(x^2 - 24 = 0\), we first need to isolate \(x^2\). We do this by adding 24 to both sides of the equation, resulting in \(x^2 = 24\).

Step 3: Solve for \(x\)

Since we have \(x^2 = 24\), we take the square root of both sides to solve for \(x\). This gives us two possible solutions: \(x = \sqrt{24}\) and \(x = -\sqrt{24}\).

Step 4: Simplify the Solutions

The square root of 24 can be simplified. \(\sqrt{24}\) is equivalent to \(\sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\). Therefore, the solutions are \(x = 2\sqrt{6}\) and \(x = -2\sqrt{6}\).

Key Concepts

Polynomial

Polynomials are mathematical expressions involving a sum of powers of variables. Each term in a polynomial can contain a variable raised to an exponent and a coefficient, which is a constant number multiplying the variable. For example, in the expression \(x^2 - 24\), we have a polynomial of degree 2. The degree of a polynomial is determined by the highest exponent of the variable. Here, \(x^2\) is the highest power, so the degree is 2, making it a quadratic polynomial.Key characteristics of polynomials:

• They include terms with non-negative integer exponents.
• They are classified by their degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
• The coefficients can be any real numbers.

Understanding polynomials is crucial for solving equations since different rules apply based on the degree of the polynomial. Quadratics, like \(x^2 - 24 = 0\), often have special methods of solution.

Square Root

The square root is a fundamental concept in mathematics. Taking the square root of a number means finding a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, since \(3 \times 3 = 9\). In the context of solving quadratic equations, we often encounter square roots when isolating the variable term.When taking the square root in equations like \(x^2 = 24\), we obtain two solutions:

• One positive: \(x = \sqrt{24}\)
• One negative: \(x = -\sqrt{24}\)

The reason for the two solutions is that both positive and negative numbers, when squared, yield the same result. Simplifying the square root involves breaking it down into simpler components, as shown with \(\sqrt{24} = 2\sqrt{6}\). Simplification helps to represent numbers in their simplest, most understandable form.

Solving Equations

Solving equations is the process of finding values for the variables that make the equation true. For quadratic equations like \(x^2 - 24 = 0\), the main objective is to determine the values of \(x\) that satisfy the equation.Key steps in solving quadratic equations:

• Isolate the quadratic term, here resulting in \(x^2 = 24\).
• Apply the square root to both sides to solve for \(x\), which gives the possible solutions.
• Simplify any square roots to provide a clear result.

For our example, we determined that \(x = 2\sqrt{6}\) and \(x = -2\sqrt{6}\). The solutions cover both possible positive and negative roots, emphasizing the need to account for all potential answers in equations involving squares.