Solving For Q: Find Values For A & B In Equation

by Alex Johnson 49 views

√{3q} = 27$

Let's break down how to find the correct values for A and B in the equation $3

√{3q} = 27$. This problem involves solving for the variable q in a multi-step equation that includes a square root. Understanding each step is crucial not only for this specific problem but also for tackling similar algebraic challenges. We'll walk through each step, explain the reasoning behind it, and pinpoint exactly how to arrive at the values for A and B. So, let’s dive in and make sure every part of the solution is crystal clear!

Step-by-Step Solution

The initial equation we're given is:

$3

√{3q} = 27$

The goal here is to isolate q on one side of the equation. To do this, we'll follow a series of algebraic manipulations, each designed to simplify the equation until q stands alone.

Step 1: Isolate the Square Root

The first thing we need to do is get the square root term by itself. Notice that the square root, $

√{3q}$, is being multiplied by 3. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by 3:

$3

√{3q} / 3 = 27 / 3$

This simplifies to:

$

√{3q} = 9$

This step is crucial because it sets us up to eliminate the square root in the next step. By isolating the square root, we make it easier to apply the inverse operation.

Step 2: Eliminate the Square Root

Now that we have the square root isolated, we can eliminate it by squaring both sides of the equation. Remember, whatever operation you perform on one side of the equation, you must perform on the other to maintain equality. Squaring both sides gives us:

$(

√{3q})^2 = 9^2$

The square root and the square cancel each other out on the left side, leaving us with:

3q=813q = 81

Here, 929^2 equals 81. This step is a key moment in the solution, and we can see that the value of A in Jacob's work corresponds to this step. Therefore:

A = 81

So, we've found the first value we were looking for! Now let’s move on to the final step to find B.

Step 3: Solve for q

We're almost there! We now have the equation:

3q=813q = 81

To isolate q, we again use the inverse operation. Since q is being multiplied by 3, we divide both sides of the equation by 3:

3q/3=81/33q / 3 = 81 / 3

This simplifies to:

q=27q = 27

And there we have it! The value of q is 27. Looking back at Jacob's work, B represents the final value of q. Thus:

B = 27

Correct Values for A and B

So, to recap, we've found that:

  • A = 81
  • B = 27

These are the correct values for A and B that satisfy the original equation and the steps Jacob was using to solve it. Understanding the order of operations and how to apply inverse operations is essential for solving algebraic equations like this one.

Common Mistakes and How to Avoid Them

When solving equations involving square roots, there are a few common pitfalls that students often encounter. Let's discuss these mistakes and how to steer clear of them.

Forgetting to Square Both Sides

One frequent mistake is squaring only one side of the equation when trying to eliminate the square root. Remember, to maintain the balance of the equation, any operation performed on one side must be performed on the other. For instance, if you have $

√{3q} = 9$, you must square both sides $(

√{3q})^2 = 9^2$ to correctly eliminate the square root. Squaring only the side with the square root would lead to an incorrect result.

Incorrect Order of Operations

Another common error is not following the correct order of operations. In our case, it’s crucial to isolate the square root before squaring. If you were to square the original equation $3

√{3q} = 27$ directly without first dividing by 3, you would end up with a more complex equation that is harder to solve. Always isolate the radical first to simplify the process.

Arithmetic Errors

Simple arithmetic mistakes can also throw off your solution. For example, miscalculating 929^2 or incorrectly dividing 81 by 3 can lead to the wrong values for A and B. Always double-check your calculations, especially in multi-step problems where an early error can propagate through the rest of the solution.

Not Checking the Solution

Finally, it’s always a good practice to check your solution by plugging the value of q back into the original equation. This ensures that your solution is correct and satisfies the equation. For instance, if we plug q = 27 back into $3

√{3q} = 27$, we get $3

√{3 * 27} = 27$, which simplifies to $3

√{81} = 27$, and further to 3βˆ—9=273 * 9 = 27, which is true. This confirms that our solution is correct.

By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving algebraic equations involving square roots.

Real-World Applications of Solving Equations with Square Roots

Solving equations with square roots isn't just an abstract mathematical exercise; it has numerous practical applications in various fields. Understanding these real-world uses can make the math feel more relevant and engaging. Let's explore a few examples.

Physics: Calculating Velocity and Speed

In physics, square roots often appear when calculating velocity or speed. For instance, the formula for the velocity (v) of an object under constant acceleration (a) over a distance (d) can be expressed as:

v=√2adv = √{2ad}

If you need to find the acceleration given the velocity and distance, you would have to solve an equation involving a square root. This is crucial in fields like mechanics and aerospace engineering, where understanding the motion of objects is essential.

Engineering: Structural Design

Engineers use equations with square roots in structural design to calculate the strength and stability of structures. For example, the natural frequency (f) of a vibrating string or beam is related to its physical properties (like tension T, mass per unit length ΞΌ, and length L) by equations that include square roots:

f=(1/2L)βˆ—βˆšT/ΞΌf = (1 / 2L) * √{T / ΞΌ}

When designing bridges, buildings, or other structures, engineers need to ensure that these frequencies are within acceptable limits to prevent resonance and potential collapse. Solving for variables within these equations is a common task.

Finance: Compound Interest

In finance, square roots can be used to calculate growth rates and investment returns. The formula for compound interest often involves exponents and roots. For example, if you want to find the interest rate needed to achieve a certain return on investment over a specific period, you might encounter an equation that requires solving for a square root or other radicals.

Computer Graphics and Game Development

Square roots are fundamental in computer graphics and game development for calculations involving distances and scaling. The Euclidean distance between two points in a 2D or 3D space is calculated using the square root of the sum of the squares of the differences in coordinates. This is used in collision detection, character movement, and rendering graphics.

Navigation and GPS Systems

Global Positioning Systems (GPS) rely on calculating distances between satellites and receivers on the ground. These calculations often involve square roots, as the distance is determined using the Pythagorean theorem in three dimensions. Solving these equations accurately is critical for precise navigation.

These examples illustrate just a few of the many ways equations with square roots are used in real-world applications. From designing safer structures to calculating investment returns, the ability to solve these equations is a valuable skill in numerous professions.

In conclusion, solving the equation $3

√{3q} = 27$ to find the values of A and B is a great exercise in algebraic manipulation. By isolating the square root, squaring both sides, and then solving for q, we found that A = 81 and B = 27. Remember to avoid common mistakes like squaring only one side or miscalculating arithmetic. Understanding these steps is not only crucial for academic success but also provides a foundation for real-world applications in fields like physics, engineering, and finance.

For further exploration of algebraic equations and square roots, consider visiting Khan Academy's Algebra I section.