Solve the Rational Equation for x square root of x+8 = square root of 3x+8
The problem is asking to find the value(s) of the variable \( x \) for which the given equation is true. The equation presented is an equation involving radicals (square roots) with the variable within the radicand (the expression under the square root symbol). The challenge is to solve for \( x \) in the equation that features a square root containing a linear expression with \( x \) on both sides of the equation. Typically, such a problem would be approached by isolating square roots on one side of the equation and then squaring both sides to eliminate the square roots, followed by simplifying and solving what should then be a polynomial equation.
$\sqrt{x + 8} = \sqrt{3 x + 8}$
Answer
Solution:
Step 1:
Eliminate the square root on the left by squaring both sides of the equation. $(\sqrt{x + 8})^{2} = (\sqrt{3x + 8})^{2}$
Step 2:
Expand both sides of the equation.
Step 2.1:
Express $\sqrt{x + 8}$ as $(x + 8)^{\frac{1}{2}}$. $(x + 8)^{\frac{1}{2} \cdot 2} = (\sqrt{3x + 8})^{2}$
Step 2.2:
Resolve the left side.
Step 2.2.1:
Simplify $(x + 8)^{\frac{1}{2} \cdot 2}$.
Step 2.2.1.1:
Utilize the exponent multiplication rule, $(a^{m})^{n} = a^{m \cdot n}$. $(x + 8)^{\frac{1}{2} \cdot 2} = (\sqrt{3x + 8})^{2}$
Step 2.2.1.1.2:
Reduce the fraction in the exponent. $(x + 8)^{1} = (\sqrt{3x + 8})^{2}$
Step 2.2.1.2:
Simplify to $x + 8 = (\sqrt{3x + 8})^{2}$.
Step 2.3:
Simplify the right side.
Step 2.3.1:
Express $(\sqrt{3x + 8})^{2}$ as $3x + 8$.
Step 2.3.1.1:
Apply the exponent rule to get $x + 8 = (3x + 8)^{1}$.
Step 2.3.1.2:
Simplify to $x + 8 = 3x + 8$.
Step 3:
Isolate $x$.
Step 3.1:
Transfer all $x$ terms to one side. $x + 8 - 3x = 8$
Step 3.2:
Move constants to the opposite side. $-2x + 8 = 8$
Step 3.3:
Divide to solve for $x$.
Step 3.3.1:
Divide $-2x = 0$ by $-2$. $\frac{-2x}{-2} = \frac{0}{-2}$
Step 3.3.2:
Cancel out $-2$. $x = \frac{0}{-2}$
Step 3.3.3:
Finalize with $x = 0$.
Knowledge Notes:
Square Roots and Exponents: The square root of a number is equivalent to raising that number to the power of $\frac{1}{2}$. When you square a square root, the operations cancel each other out, leaving you with the original number under the square root.
Squaring Both Sides: When you have an equation with square roots, you can square both sides to eliminate the square roots. However, be cautious as squaring can introduce extraneous solutions.
Simplifying Exponents: When you have an exponent raised to another exponent, you multiply the exponents together. This is known as the power rule: $(a^{m})^{n} = a^{m \cdot n}$.
Isolating Variables: To solve for a variable, you need to get the variable on one side of the equation and the constants on the other. This often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
Checking Solutions: After solving an equation, especially one involving square roots, it's important to check that your solution does not result in taking the square root of a negative number (if dealing with real numbers) and that it satisfies the original equation.
