Solve the Rational Equation for x square root of x+15=5+ square root of x

Solution:

Step 1:

Square both sides to eliminate the square root. $(\sqrt{x + 15})^2 = (5 + \sqrt{x})^2$

Step 2:

Simplify the equation.

Step 2.1:

Express $\sqrt{x + 15}$ as $(x + 15)^{\frac{1}{2}}$. $(x + 15)^{\frac{1}{2} \cdot 2} = (5 + \sqrt{x})^2$

Step 2.2:

Simplify the left side.

Step 2.2.1:

Simplify $(x + 15)^{\frac{1}{2} \cdot 2}$.

Step 2.2.1.1:

Multiply the exponents. $(x + 15)^{1} = (5 + \sqrt{x})^2$

Step 2.2.1.2:

The exponent 1 does not change the base. $x + 15 = (5 + \sqrt{x})^2$

Step 2.3:

Simplify the right side.

Step 2.3.1:

Expand $(5 + \sqrt{x})^2$.

Step 2.3.1.1:

Use the FOIL method to expand. $x + 15 = (5 + \sqrt{x})(5 + \sqrt{x})$

Step 2.3.1.2:

Distribute each term. $x + 15 = 25 + 5\sqrt{x} + 5\sqrt{x} + (\sqrt{x})^2$

Step 2.3.1.3:

Combine like terms. $x + 15 = 25 + 10\sqrt{x} + x$

Step 3:

Isolate $10\sqrt{x}$.

Step 3.1:

Rearrange the equation. $25 + 10\sqrt{x} + x = x + 15$

Step 3.2:

Move non-radical terms to the other side.

Step 3.2.1:

Subtract 25 from both sides. $10\sqrt{x} + x = x + 15 - 25$

Step 3.2.2:

Subtract x from both sides. $10\sqrt{x} = x + 15 - 25 - x$

Step 3.2.3:

Combine terms. $10\sqrt{x} = -10$

Step 4:

Square both sides to eliminate the square root. $(10\sqrt{x})^2 = (-10)^2$

Step 5:

Simplify both sides.

Step 5.1:

Express $\sqrt{x}$ as $x^{\frac{1}{2}}$. $(10x^{\frac{1}{2}})^2 = (-10)^2$

Step 5.2:

Simplify the left side.

Step 5.2.1:

Apply the power rule. $100x^{\frac{1}{2} \cdot 2} = (-10)^2$

Step 5.2.1.1:

Simplify the exponent. $100x = (-10)^2$

Step 5.3:

Simplify the right side.

Step 5.3.1:

Square -10. $100x = 100$

Step 6:

Divide by 100 to solve for x.

Step 6.1:

Divide each side by 100. $\frac{100x}{100} = \frac{100}{100}$

Step 6.2:

Simplify the left side. $x = \frac{100}{100}$

Step 6.3:

Simplify the right side. $x = 1$

Step 7:

Check the solution in the original equation. No solution is excluded, $x = 1$ is valid.

Knowledge Notes:

1. Square Roots and Exponents: Squaring both sides of an equation is a common technique to eliminate square roots. Remember that $(\sqrt{a})^2 = a$ and $(a^{\frac{1}{2}})^2 = a$.

2. Simplification: After squaring, it's important to simplify each side of the equation separately, combining like terms and using the distributive property (FOIL method) when necessary.

3. Power Rule: When raising a power to a power, you multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

4. Combining Like Terms: When simplifying expressions, add or subtract like terms to consolidate the equation.

5. Isolating Variables: To solve for a variable, get all terms with that variable on one side and constants on the other. Then isolate the variable by performing inverse operations.

6. Checking Solutions: Always plug the solution back into the original equation to ensure it does not result in an undefined expression or a contradiction.