Problem 44 - Algebra 2

Question

Simplify. \((\sqrt{11}-\sqrt{2})^{2}\)

Step-by-Step Solution

Answer

\(13 - 2\sqrt{22}\)

Step 1: Write the Expression

The expression given is \((\sqrt{11}-\sqrt{2})^{2}\). Our goal is to simplify this expression.

Step 2: Apply the Binomial Square Formula

The binomial square formula is \((a-b)^{2} = a^{2} - 2ab + b^{2}\). For the expression \((\sqrt{11}-\sqrt{2})^{2}\), we identify \(a\) as \(\sqrt{11}\) and \(b\) as \(\sqrt{2}\).

Step 3: Compute Each Square Term

Calculate \(a^{2}\) and \(b^{2}\) separately. So, \((\sqrt{11})^{2} = 11\) and \((\sqrt{2})^{2} = 2\).

Step 4: Compute the Product of the Terms

Calculate \(2ab\), which becomes \(2\times\sqrt{11}\times\sqrt{2} = 2\times\sqrt{22} = 2\sqrt{22}\).

Step 5: Substitute Back into the Formula

Substitute the values back into the formula: \(a^{2} - 2ab + b^{2} = 11 - 2\sqrt{22} + 2\).

Step 6: Simplify the Expression

Add the constant terms together: \(11 + 2 = 13\). Thus, the expression simplifies to \(13 - 2\sqrt{22}\).

Key Concepts

Binomial Square Formula

The Binomial Square Formula is a helpful tool in algebra for simplifying expressions involving the square of a binomial. A binomial expression is a sum or difference involving two terms, expressed as either \((a + b)^2\) or \((a - b)^2\). In our example, the expression \((\sqrt{11} - \sqrt{2})^2\) fits into the latter, where \(a\) is \(\sqrt{11}\) and \(b\) is \(\sqrt{2}\).

The formula for expanding the square of a binomial is:

• \((a \pm b)^2 = a^2 \pm 2ab + b^2\)

In this formula, the symbol \(±\) indicates that it applies to both the sum \((a + b)^2\) and the difference \((a - b)^2\). The formula helps us by providing a structured way to expand and simplify such expressions. It transforms a potentially complicated expression into simpler, more manageable parts, making it easier to compute squares and products of terms.

In our example, by applying the formula, we convert \((\sqrt{11} - \sqrt{2})^2\) into \(11 - 2\sqrt{22} + 2\). This makes simplifying the expression straightforward and ensures clarity in your algebraic work.

Radicals

Understanding radicals is key to solving algebraic expressions that involve square roots, like the expression in our problem. A radical \(\sqrt{n}\) signifies the square root of \(n\), which is a number that, when multiplied by itself, gives \(n\). In simpler terms, if \(x^2 = n\), then \(x = \sqrt{n}\).

Radicals are essential in breaking down complex numbers and solving equations. In our example, the radicals \(\sqrt{11}\) and \(\sqrt{2}\) appear in the expression \((\sqrt{11} - \sqrt{2})^2\). To deal with these, we must understand two properties:

• Simplifying Radicals: Simplifying involves making the radical expression as simple as possible. For perfect square numbers (like \(4, 9, 16\)), the radical simplifies to their base numbers (\(\sqrt{4} = 2\), \(\sqrt{9} = 3\), etc.).
• Multiplication of Radicals: When multiplying radicals \(\sqrt{a} \times \sqrt{b}\), the result is \(\sqrt{a \times b}\). This is used in our solution to find \(2\sqrt{22}\) when calculating the \(2ab\) term in the binomial square formula.

Mastering these basics will help you confidently manipulate radicals in algebraic expressions.

Quadratic Expression

Quadratic expressions are significant in algebra for modeling various real-world situations. These expressions typically take the form of \(ax^2 + bx + c\), though our example shows a modification involving radicals rather than variables. Our expression \((\sqrt{11} - \sqrt{2})^2\) can be viewed as a specialized case where constants and radicals replace variables.

Dealing with these expressions often involves expanding and simplifying them to establish clearer insights or reach solutions. The expansion via the binomial square formula transforms these quadratic forms into a simplified structure, revealing any inherent symmetries or simplifications.

Understanding quadratic expressions is crucial:

• Simplification: Adjust, condense, or extend expressions to make them easier to solve.
• Standard Forms: Recognizing and using the standard form, even when variables aren't present, will aid in solving these expressions.

The knowledge of quadratics extends past basic equations to more complex expressions where simplification or factorization might be necessary, just as seen in the algebraic manipulation of our exercise.