Problem 40 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ 4 x^{2}=3 $$

Step-by-Step Solution

Answer

The solution is \(x = \pm\frac{\sqrt{3}}{2}\).

Step 1: Set Up the Equation

We start with the equation given in the problem: \(4x^2 = 3\). Our goal is to solve for \(x\).

Step 2: Divide by 4

To isolate \(x^2\), we divide both sides of the equation by 4. This results in \(x^2 = \frac{3}{4}\).

Step 3: Take the Square Root

To solve for \(x\), take the square root of both sides. Remember that taking the square root introduces both a positive and a negative solution: \(x = \pm\sqrt{\frac{3}{4}}\).

Step 4: Simplify the Square Root

Simplify \(\sqrt{\frac{3}{4}}\) by taking the square root of the numerator and the denominator separately: \(x = \pm\frac{\sqrt{3}}{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are expressions that involve terms up to the second power, typically written in the form \( ax^2 + bx + c = 0 \). The equation you've seen, \(4x^2 = 3 \), is a special type of quadratic equation where the linear term (the one with \(x\)) and the constant are initially zero. This makes the process less complex as we only focus on the \(x^2\) term.

Quadratic equations are fundamental in algebra, and they can model a range of real-world scenarios, from physics to finance. Solving these equations can help predict motion, optimize designs, or even let you get a discount on a mortgage!

To solve quadratic equations, we often rely on different methods such as:

• Factoring
• Completing the square
• Using the quadratic formula
• Graphing

For the simpler quadratic equation like \(4x^2 = 3\), a straightforward method involving simplification and taking square roots is often used.

Solving Equations

Solving equations means finding the value of the variable that makes the equation true. In our equation, \(4x^2 = 3\), we want to determine the value of \(x\).

We begin by isolating \(x^2\) from other terms, helping us to solve the equation step by step. This process involves logical operations such as division and simplification:

• *Isolating Terms:* Divide the entire equation by 4 to make the \(x^2\) term standalone (\(x^2 = \frac{3}{4}\)).
• *Balancing the Equation:* Keep the equation balanced by doing the same operation to both sides. This is crucial to maintain equality.

Each step you compute should be verified by plugging the values back into the original equation. This not only ensures accuracy but also strengthens the understanding of equation solving as a concept.

Square Root Property

The square root property is a simplified method for solving equations that involve a squared term only. When you have an equation in the form \( x^2 = a \), you can solve for \(x\) by taking the square root of both sides.

This property points out that every number has two square roots: a positive and a negative one. So, for \(x^2 = \frac{3}{4}\), we apply the square root property to get:

• \( x = \pm \sqrt{\frac{3}{4}} \)

Additionally, you simplify the expression further. Breaking down the square root into its numerator and denominator can often make the solution more elegant:

• \( x = \pm \frac{\sqrt{3}}{2} \)

Using the square root property is an efficient tactic for solving equations when working with perfect squares or when a direct factor isn't obvious. With this property, solving quadratic expressions becomes intuitive and often less error-prone, especially when you remember that the square root symbol denotes both possible roots.