Problem 41 - Elementary Algebra

Question

Solve by extracting the roots. $$ 16 x 2-27=0 $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Square Term

Start by isolating the term containing the square. The equation is \( 16x^2 - 27 = 0 \). Add 27 to both sides to obtain \( 16x^2 = 27 \).

Step 2: Divide by the Coefficient of the Square Term

Divide both sides of the equation by 16 to solve for \( x^2 \): \( x^2 = \frac{27}{16} \).

Step 3: Extract the Square Roots

Take the square root of both sides to solve for \( x \). Remember that the square root operation produces two solutions: \[ x = \pm \sqrt{\frac{27}{16}} \].

Step 4: Simplify the Square Root

Simplify the fraction inside the square root. Since \( \sqrt{\frac{27}{16}} = \frac{\sqrt{27}}{\sqrt{16}} \) and \( \sqrt{16} = 4 \), we have: \[ x = \pm \frac{\sqrt{27}}{4} \].

Step 5: Simplify the Radical Expression

Recognize that \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \). Substitute back to get \[ x = \pm \frac{3\sqrt{3}}{4} \].

Key Concepts

Extracting Roots

When solving quadratic equations such as \( 16x^2 - 27 = 0 \) by extracting roots, the key is to focus on isolating the term with the square. Here's how you do it:

• First, ensure that the square term is alone on one side of the equation. In our example, we have \( 16x^2 = 27 \).
• Next, you need to eliminate any coefficients attached to the square term. This involves dividing both sides of the equation by that coefficient. So, we divide both sides by 16, leaving us with \( x^2 = \frac{27}{16} \).
• The final step to extract the roots is to take the square root of both sides. This will yield two potential solutions for \( x \): \( x = \pm \sqrt{\frac{27}{16}} \).

Remember, taking the square root gives both a positive and negative solution. So don't forget to include the \( \pm \) symbol in your solutions.

Simplifying Radicals

Now that we have the roots as \( x = \pm \sqrt{\frac{27}{16}} \), it's time to simplify the radicals.

• The process of simplifying radicals involves breaking down the square root into more manageable pieces.
• First, deal with the denominator: \( \sqrt{16} = 4 \). So, this part becomes \( \frac{\sqrt{27}}{4} \).
• Next, simplify \( \sqrt{27} \). Notice that 27 can be factored as 9 times 3. This gives us \( \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \).

After breaking it down, combine everything to get \( x = \pm \frac{3\sqrt{3}}{4} \). By simplifying radicals, you make the expression neater and often easier to work with.

Solving Equations

Solving quadratic equations can seem daunting, but with a systematic approach, it becomes manageable.

• Start by identifying the type of equation you're dealing with. For quadratics, look for terms involving squared variables, like \( x^2 \).
• The goal is to isolate this term and solve for the variable. This often means rearranging the equation, like moving constants to the other side.
• Once isolated, use operations such as extracting roots (taking square roots) to find potential solutions.

But solving doesn't stop just at finding \( x \). You must also verify that these solutions satisfy the original equation. Always substitute back to check. This ensures accuracy and that no mistakes were made during simplification or extraction.