Algebra - "Solve for x" Challenge¶

Problem Statement¶

Solve for \(x\):

\[\sqrt{x} + \sqrt{2} = \sqrt{32}\]

Step-by-Step Solution¶

We are using the isolation method, which involves moving all constants to the right-hand side (RHS), and isolating the variable term on the left-hand side (LHS). Let's proceed carefully.

Step 1: Start with the given equation¶

\[ \sqrt{x} + \sqrt{2} = \sqrt{32} \]

Step 2: Subtract \(\sqrt{2}\) from both sides¶

To isolate the variable \(\sqrt{x}\), we subtract \(\sqrt{2}\) from both sides of the equation. This uses the Subtraction Property of Equality, which states that for any numbers \(a\), \(b\), and \(c\), if \(a = b\), then \(a - c = b - c\). Applying this property to the equation:

\[ \sqrt{x} + \sqrt{2} - \sqrt{2} = \sqrt{32} - \sqrt{2} \]

On the left-hand side, \(\sqrt{2} - \sqrt{2} = 0\), so we are left with:

\[ \sqrt{x} = \sqrt{32} - \sqrt{2} \]

Step 3: Simplify the right-hand side (RHS)¶

Now, simplify the terms on the RHS. We first simplify \(\sqrt{32}\) using the Product Property of Square Roots, which states \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\):

\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

So the equation becomes:

\[ \sqrt{x} = 4\sqrt{2} - \sqrt{2} \]

Step 4: Factor out \(\sqrt{2}\)¶

Now, factor out \(\sqrt{2}\) from the RHS using the Distributive Property:

\[ \sqrt{x} = (4 - 1)\sqrt{2} \]

Simplifying the factor: [ \sqrt{x} = 3\sqrt{2} ]

Step 5: Square both sides¶

To eliminate the square root on the LHS, square both sides of the equation. This uses the Square Property of Equality, which states that if \(a = b\), then \(a^2 = b^2\):

\[ (\sqrt{x})^2 = (3\sqrt{2})^2 \]

On the left-hand side:

\[ (\sqrt{x})^2 = x \]

💡: Why does \((\sqrt{x})^2 \) resolve to \(x\)?:

Start with the initial expression:

\[(\sqrt{x})^2\]

Re-express \(\sqrt{x}\) as \(x^{1/2}\) (translate \(\sqrt{x}\) to its rational exponent equivalence):

\[(\sqrt{x})^2 = (x^{1/2})^2\]

Simplify \((x^{1/2})^2\) using the power rule of exponents:

\[(x^{1/2})^2 = x^{(1/2) \cdot 2}\]

Since \((1/2) \cdot 2 = 1\):

\[x^{(1/2) \cdot 2} = x^1 = x\]

On the right-hand side, use the Product Property of Exponents, which states \((ab)^2 = a^2 \times b^2\):

\[ (3\sqrt{2})^2 = 3^2 \times (\sqrt{2})^2 = 9 \times 2 = 18 \]

Thus, the equation becomes:

\[ x = 18 \]

Step 6: Check the solution¶

To verify, substitute \(x = 18\) back into the original equation:

\[ \sqrt{18} + \sqrt{2} = \sqrt{32} \]

Simplify \(\sqrt{18}\):

\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]

So the left-hand side becomes:

\[ 3\sqrt{2} + \sqrt{2} = 4\sqrt{2} \]

The right-hand side is:

\[ \sqrt{32} = 4\sqrt{2} \]

Since both sides are equal, \(x = 18\) is correct.

Solution:¶

\[ x = 18 \]

Arithmetic and Algebra Properties Used In Solving for \(x\):¶

Subtraction Property of Equality: In Step 2, \(\sqrt{2}\) was subtracted from both sides.
Product Property of Square Roots: In Step 3, \(\sqrt{32}\) was simplified.
Distributive Property: In Step 4, \(\sqrt{2}\) was factored out.
Square Property of Equality: In Step 5, both sides were squared to eliminate the square root.
Product Property of Exponents: In Step 5, \((3\sqrt{2})^2\) was expanded.