Problem 48 - Algebra 1: Concepts and Skills

Question

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ 16+x^{2}=64 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 4*\sqrt{3}\), and \(x = -4*\sqrt{3}\)

Step 1: Simplify the equation

Subtract 16 from both sides of the equation to isolate \(x^2\). This results in the equation: \(x^2 = 64 - 16 = 48\)

Step 2: Solve for 'x'

Find the square root of both sides. Remember that the square root of a number has both a positive and a negative value. Therefore, \(x = +\sqrt{48}\), and \(x = -\sqrt{48}\)

Step 3: Simplify the radical

The square root of 48 can be simplified by factoring out perfect squares. The prime factorization of 48 includes the perfect square 16. So, \(x = +\sqrt{16*3}=+\sqrt{16}*\sqrt{3}=+4*\sqrt{3}\), and \(x = -\sqrt{16*3}=-\sqrt{16}*\sqrt{3}=-4*\sqrt{3}\)

Key Concepts

Radical Expressions

When we talk about radical expressions, we mean any expression that includes a square root, cube root, or any higher-order root. These mathematical expressions make use of radicals (the symbols used to denote roots). A common form of radical expressions involves square roots.

• For example, \(\sqrt{48}\) is a radical expression because it contains a square root.
• The number inside the radical sign is called the radicand. In our case, 48 is the radicand.
• An important aspect of simplifying radical expressions involves factoring the radicand to find perfect squares.

Breaking down radical expressions makes them easier to understand and work with, especially when finding exact solutions to equations.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because \(4 \times 4 = 16\).
When solving equations, it's important to remember:

• Square roots have both positive and negative solutions, due to the property that both \((-x)(-x)\) and \((x)(x)\) produce the same result.
• Not every number is a perfect square, which means its square root can't be expressed as a whole number, like \(\sqrt{48}\).

Finding the square root is often the key step in solving quadratic equations, as demonstrated in the example equation \(x^2 = 48\) from above. By taking the square root of both sides, we found two possible answers for \(x\).

Factoring Perfect Squares

Factoring perfect squares is a method used to simplify expressions involving square roots. This involves recognizing and extracting the largest perfect square that is a factor of the number. For instance:

• In our example expression \(x^2 = 48\), we identify 16 as the largest perfect square factor of 48 because \(48 = 16 \times 3\).
• The square root of \(48\) can thus be expressed as \(\sqrt{16} \times \sqrt{3} = 4\sqrt{3}\), which simplifies the radical expression.
• This makes calculations easier and is especially useful in solutions that require precise representations, like in problems with quadratic equations.

This simplification lets us express answers using radical expressions in a cleaner form, often necessary when we can't simplify a square root to an integer.