Problem 51 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 8 x^{2}-64=0 $$

Step-by-Step Solution

Answer

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

Step 1: Set the Equation to Zero

The given equation is \(8x^2 - 64 = 0\). Since it is already set to zero, we can start solving for \(x\).

Step 2: Isolate the Squared Term

Add 64 to both sides of the equation to isolate the squared term:\[8x^2 = 64\]

Step 3: Divide by the Coefficient of the Squared Term

Divide both sides by 8 to solve for \(x^2\):\[x^2 = 8\]

Step 4: Solve for x

Take the square root of both sides to solve for \(x\). Remember to consider both the positive and negative roots:\[x = \pm \sqrt{8}\]Since \(\sqrt{8}\) can be further simplified, you end up with:\[x = \pm 2\sqrt{2}\]

Step 5: State the Solutions

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

Key Concepts

Real Solutions

When we talk about real solutions in quadratic equations, we're referring to the values of the variable that satisfy the equation and are real numbers. These solutions can be integers or irrational numbers but cannot be imaginary. Quadratic equations typically have two solutions due to the squared term.

• Both solutions of an equation are real when the discriminant, a part of the quadratic formula, is greater than or equal to zero.
• For our equation, if the squared term equals a positive number, we get real solutions, as both positive and negative roots exist.

In the equation \(8x^2 - 64 = 0\), we isolated \(x^2 = 8\), which yields real solutions \(x = \pm 2\sqrt{2}\). This is because \(\sqrt{8}\) is a real number, and the square roots provide both positive and negative values, such as \(2\sqrt{2}\) and \(-2\sqrt{2}\). It is vital to consider both the plus and minus roots in order to capture all possible real solutions.

Solving Algebraic Equations

Solving algebraic equations involves finding all possible values of the variable that make the equation true. Quadratic equations like the one we have can usually be solved through several methods:

• Factoring: If the equation is factorable, it can be rewritten as a product of simpler expressions set to zero.
• Using the quadratic formula: This is a general method that works for any quadratic equation.
• Completing the square: This method helps in converting a quadratic equation into a perfect square trinomial.

In our example with \(8x^2 - 64 = 0\), we took a straightforward approach by isolating \(x^2\) and using square roots to find \(x\). This particular quadratic didn't require factoring or the quadratic formula due to its simplicity in structure.

Square Roots

Taking square roots is a vital part of solving certain quadratic equations, especially when the equation is already simplified to a form like \(x^2 = n\). Here, finding \(x\) involves determining both the positive and negative roots:

• The principal square root is the positive root, \(\sqrt{n}\).
• The negative root completes the solutions, \(-\sqrt{n}\).

In solving \(x^2 = 8\), for example, finding \(\sqrt{8}\) involves recognizing that this is \(2\sqrt{2}\), as 8 is equal to \(4 \times 2\) and the square root of 4 is 2. Thus, the solutions \(x = 2\sqrt{2}\) and \(x = -2\sqrt{2}\) accurately represent both possibilities. Understanding how to break down and simplify square roots is crucial for accurately solving such equations.