Problem 54 - Algebra 2

Question

Simplify. $$ (5-\sqrt{3})^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(28 - 10\sqrt{3}\).

Step 1: Understand the Expression

The expression given is \((5-\sqrt{3})^2\). This indicates that we must square the binomial \((5-\sqrt{3})\). To simplify, we will apply the formula for squaring a binomial.

Step 2: Apply the Binomial Square Formula

The formula for squaring a binomial \((a-b)^2\) is \(a^2 - 2ab + b^2\). Here, \(a = 5\) and \(b = \sqrt{3}\). Substitute these values into the formula.

Step 3: Calculate Each Term

Compute each part of the formula:1. \(a^2 = 5^2 = 25\)2. \(-2ab = -2 \times 5 \times \sqrt{3} = -10\sqrt{3}\)3. \(b^2 = (\sqrt{3})^2 = 3\).

Step 4: Sum the Terms

Now, combine all the calculated terms:\(a^2 - 2ab + b^2 = 25 - 10\sqrt{3} + 3\). Add the numerical constants together:\(25 + 3 = 28\). Thus, the simplified expression is:\(28 - 10\sqrt{3}\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool that helps in expanding binomials raised to a power. A binomial is simply an algebraic expression containing two terms, for example, \((a + b)\) or \((a - b)\). The theorem provides a general formula for expanding any power of a binomial, making the process of expansion simpler and more organized.

For squaring a binomial, like our example \((5 - \sqrt{3})^2\), we use a specific case of the binomial theorem called the square of a binomial formula. This formula is \((a - b)^2 = a^2 - 2ab + b^2\), where the first term is squared, two times the product of the two terms is subtracted, and the second term is also squared.

In practice, this means if we have a binomial \((x - y)^2\), we would calculate the expression as:

• \(x^2\)
• \(-2xy\)
• \(y^2\)

Then, we would sum these to get the expanded form. This method helps to manage the complexity when dealing with binomials raised to a power, whether whole numbers or algebraic expressions.

Square Root Simplification

Simplifying square roots is an essential skill in algebra, especially when dealing with non-perfect squares. The square root simplification process involves rewriting square roots in their simplest form. This means reducing the expression inside the square root as much as possible to make calculations easier.

In our case, \(\sqrt{3}\) is already in its simplest form, as 3 is a prime number and does not have a pair of factors to further simplify it. However, when square roots are involved in operations such as multiplication or squaring, it’s important to leverage properties of square roots. For example, \((\sqrt{3})^2\) simplifies to 3 because squaring a square root removes the radical.

To simplify square roots effectively, remember:

• If the radicand (the number inside the square root) can be factored into a perfect square, do so to simplify.
• Recall that \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), allowing multiplication of numbers under a square root.

Mastering these techniques can significantly ease dealing with square roots in algebraic expressions.

Polynomial Expression Simplification

Polynomial expression simplification is the process of condensing a polynomial to its simplest form. Combining like terms, performing arithmetic operations, and applying algebraic identities are essential steps in this process.

In our exercise where we simplify \((5 - \sqrt{3})^2\), simplification begins with applying the binomial square formula to break down and expand the polynomial. Once expanded, it’s crucial to calculate each term precisely and combine like terms.

Here's a brief outline of how to simplify such polynomial expressions:

• Use algebraic identities to expand the expression, like \((a-b)^2 = a^2 - 2ab + b^2\) used here.
• Calculate the value of each term in the expression separately.
• Sum like terms to reach the simplest form of the polynomial.

Simplifying polynomials reduces complexity in subsequent calculations and makes equations more manageable, aiding in further algebraic manipulations or solving equations.