Problem 291 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((5-\sqrt{10})^{2}\) (b) \((8+3 \sqrt{2})^{2}\)

Step-by-Step Solution

Answer

(a) 35 - 10\sqrt{10}, (b) 82 + 48\sqrt{2}

Step 1: Expand the expression \( (a - b)^2 \)

Use the formula \( (a - b)^2 = a^2 - 2ab + b^2 \) on the expression \( (5 - \sqrt{10})^2 \). Here, \(a = 5\) and \(b = \sqrt{10}\).

Step 2: Apply the formula

Plug values into the formula: \(5^2 - 2(5)(\sqrt{10}) + (\sqrt{10})^2 \).

Step 3: Simplify each term

Calculate each term individually: \(5^2 = 25\), \(-2(5)(\sqrt{10}) = -10\sqrt{10}\), and \((\sqrt{10})^2 = 10\).

Step 4: Combine terms

Combine the simplified terms to get \(25 - 10\sqrt{10} + 10 = 35 - 10\sqrt{10}\).

Step 5: Expand the expression \( (a + b)^2 \)

Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) on the expression \( (8 + 3\sqrt{2})^2 \). Here, \ a = 8 \ and \ b = 3\sqrt{2}\.

Step 6: Apply the formula

Plug values into the formula: \(8^2 + 2(8)(3\sqrt{2}) + (3\sqrt{2})^2 \).

Step 7: Simplify each term

Calculate each term individually: \(8^2 = 64\), \(2(8)(3\sqrt{2}) = 48\sqrt{2}\), and \((3\sqrt{2})^2 = 18\).

Step 8: Combine terms

Combine the simplified terms to get \(64 + 48\sqrt{2} + 18 = 82 + 48\sqrt{2}\).

Key Concepts

square of a binomial

When dealing with the square of a binomial, it’s important to remember the formula \[ (a \pm b)^2 = a^2 \pm 2ab + b^2 \].
This formula works for both addition and subtraction.

For example, let’s consider the expression \((5 - \sqrt{10})^2\).
Here, a = 5 and b = \sqrt{10}\. You would first square each term individually, multiple the two terms, and then combine them according to the formula.

Simplifying using the formula, we get:
\[ (5 - \sqrt{10})^2 = 5^2 - 2(5)(\sqrt{10}) + (\sqrt{10})^2 \]
Which simplifies to:
\[ 25 - 10\sqrt{10} + 10 = 35 - 10\sqrt{10}\]
This gives us the final result of the expression.

expanding expressions

Expanding expressions is a fundamental skill in algebra, especially when working with binomials.
It allows you to transform a factored expression into a sum or difference that is easier to simplify.