Problem 23 - Elementary Algebra

Question

Solve by extracting the roots. $$ 16 x 2=9 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3}{4}\) and \(x = -\frac{3}{4}\).

Step 1: Isolate the variable term

To begin solving the equation \(16x^2 = 9\), divide both sides of the equation by 16 to isolate the term with \(x^2\). This gives us \(x^2 = \frac{9}{16}\).

Step 2: Take the square root of both sides

To solve for \(x\), take the square root of both sides of the equation \(x^2 = \frac{9}{16}\). This gives \(x = \pm \sqrt{\frac{9}{16}}\).

Step 3: Simplify the square root

Simplify \(\sqrt{\frac{9}{16}}\) by finding the square root of the numerator and the denominator separately. \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\), so \(\sqrt{\frac{9}{16}} = \frac{3}{4}\).

Step 4: State the solutions

Since \(x = \pm \sqrt{\frac{9}{16}}\), the solutions are \(x = \frac{3}{4}\) and \(x = -\frac{3}{4}\). Therefore, the equation has two solutions: \(x = \frac{3}{4}\) and \(x = -\frac{3}{4}\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra, and they have a distinct form: \( ax^2 + bx + c = 0 \). These represent parabolas when graphed. Encountering quadratic equations can seem daunting at first, but understanding their structure is key. Each term in \( ax^2 + bx + c \) plays a role in defining the curve's shape and position.
Quadratics typically yield two solutions, which can be found using several methods, such as factoring, completing the square, or the quadratic formula. The square root method particularly comes in handy when dealing with equations missing the linear \( b \) term, like in our exercise.
Here's why quadratics are important:

• They model real-world scenarios like projectile motion.
• They feature two solutions, representing two possible states or conditions.
• They create a symmetrical curve, indicative of natural phenomena.

Understanding the essence of quadratic equations equips you with the tools to approach various algebraic problems confidently.

Solving Equations

Solving equations is about finding the values that make the equation true. The key steps involve simplifying, isolating terms, and performing operations both sides equally. These steps ensure you do not disrupt the equation's balance.
The particular equation \(16x^2 = 9\) highlights the use of the square root method. Here's how you solve it:

• First, isolate the \(x^2\) term by dividing both sides by 16. This step is crucial to remove any coefficients influencing the variable directly, leading to \(x^2 = \frac{9}{16}\).
• Next, take the square root of both sides. It’s important to remember that squaring a number is a reversible operation using square roots, and both positive and negative roots must be considered.
• Finally, interpret the square root: \( \sqrt{\frac{9}{16}} = \frac{3}{4} \). Hence, \(x = \pm \frac{3}{4}\).

Following these steps helps find the potential values of \(x\) systematically, reflecting a strong grasp of algebraic principles and methods.

Simplifying Expressions

Simplifying expressions involves condensing equations into their most basic form, making them easier to work with. This often includes reducing fractions, factoring, or finding common denominators. The goal is to make calculations clearer and solve problems efficiently.
In the given problem, simplifying \(\sqrt{\frac{9}{16}}\) showcases how to handle square roots effectively. Here's the process:

• Identify separate components: Here, we recognize that both numerator (9) and denominator (16) are perfect squares.
• Calculate each square root individually: \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\).
• Combine the results: This simplification yields \(\frac{3}{4}\).

This technique of splitting square roots into their parts is particularly useful for fractions, fostering a clearer picture of the operations involved. Simplifying aids in verifying that solutions are defensible and precise, streamlining more complex mathematical processes.