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.