Problem 32 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (8 x-3)^{2}=5 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3+\sqrt{5}}{8}\) and \(x = \frac{3-\sqrt{5}}{8}\)

Step 1: Apply Square Root Property

Take the square root of both sides of the equation, while remembering it could be positive or negative root. This leads to: \(8x - 3 = \sqrt{5}\) and \(8x - 3 = -\sqrt{5}\)

Step 2: Solve for x

Now, isolate x in both equations. First, add 3 to both sides, and then, divide by 8: \(x = \frac{3+\sqrt{5}}{8}\) and \(x = \frac{3-\sqrt{5}}{8}\)

Step 3: Conclusion

The solutions of the equation \((8x -3)^2 = 5\) are \(x = \frac{3+\sqrt{5}}{8}\) and \(x = \frac{3-\sqrt{5}}{8}\)

Key Concepts

Solving Equations

Solving equations can seem intimidating, but it's all about finding the value of the variable that makes the equation true. In algebra, an equation is a statement that asserts the equality of two expressions. To solve an equation, you need to perform operations that keep the equation balanced while moving terms around to isolate the variable.
Begin by understanding what the equation is asking. Look at it as a puzzle where you know the outcome, but you need to find the missing piece - in this case, the variable's value.

• Keep both sides of the equation balanced.
• Carefully perform inverse operations to isolate the variable.
• Verify your solution by substituting the value back into the original equation.

This systematic approach ensures you maintain balance and reach a correct conclusion.

Algebra

Algebra serves as the language of mathematics. It's all about using symbols and letters to represent numbers and quantities in equations and formulas. Understanding algebra is crucial for solving a wide range of problems, from simple ones involving only basic operations to more complex equations like quadratics.
In algebra, the order of operations is key. Always follow "PEMDAS", which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. This helps in solving expressions correctly.

• Simplify expressions using arithmetic operations.
• Use variables as placeholders for unknown values.
• Identify like terms to combine them effectively.

By grasping these basic algebra concepts, solving complicated problems becomes much less daunting.

Square Root Method

The Square Root Method is a handy tool for solving certain types of equations, especially quadratic equations where one side is a square expression. It's built on the simple idea of taking the square root of both sides to simplify the equation.
When applying the square root property, remember:

• Square roots have both a positive and negative value.
• Isolate the square term on one side of the equation first.
• After taking the square root, solve the resulting linear equation to find the possible values of the variable.

Let's consider the equation \( (8x-3)^2 = 5 \). Applying the square root method means recognizing the need to take both the positive and negative roots of 5, which gives us two equations: \( 8x - 3 = \sqrt{5} \) and \( 8x - 3 = -\sqrt{5} \).
This process splits one equation into a system of simpler equations you can solve by basic algebraic operations, leading to finding all potential solutions.