Evaluate square root of 3x+14=2

Solution:

Step 1

Isolate the radical expression on one side of the equation.

Step 1.1

Subtract 14 from both sides to achieve this.$\sqrt{3x} = 2 - 14$

Step 1.2

Calculate the difference on the right side.$\sqrt{3x} = -12$

Step 2

Square both sides to eliminate the square root.$(\sqrt{3x})^2 = (-12)^2$

Step 3

Simplify the equation.

Step 3.1

Express $\sqrt{3x}$ as $(3x)^{\frac{1}{2}}$. Then square both sides.$((3x)^{\frac{1}{2}})^2 = (-12)^2$

Step 3.2

Simplify the left side by applying exponent rules.

Step 3.2.1

Apply the power of a power rule.

Step 3.2.1.1

Multiply the exponents.$(3x)^{\frac{1}{2} \cdot 2} = (-12)^2$

Step 3.2.1.1.1

Simplify the exponent by multiplying.$(3x)^{1} = (-12)^2$

Step 3.2.1.1.2

The exponent of 1 indicates the number itself.$3x = (-12)^2$

Step 3.3

Simplify the right side by squaring -12.$3x = 144$

Step 4

Divide both sides by 3 to solve for x.

Step 4.1

Divide each term by 3.$\frac{3x}{3} = \frac{144}{3}$

Step 4.2

Simplify the left side.

Step 4.2.1

Reduce the fraction by canceling out the 3s.$\frac{\cancel{3}x}{\cancel{3}} = \frac{144}{3}$

Step 4.3

Simplify the right side by dividing.$x = 48$

Step 5

Check the solution in the original equation. Since the square root of a real number cannot be negative, the equation $\sqrt{3x} + 14 = 2$ has no solution when $x = 48$.

Knowledge Notes:

To solve an equation involving a square root:

1. Isolate the square root: The first step is to have the square root on one side of the equation and all other terms on the other side.

2. Square both sides: To eliminate the square root, square both sides of the equation. Remember that squaring is the inverse operation of taking a square root.

3. Simplify: After squaring, simplify both sides of the equation. This often involves combining like terms or applying exponent rules.

4. Solve for the variable: Once simplified, solve for the variable as you would in any linear equation.

5. Check your solution: It's important to check your solution by substituting it back into the original equation. Sometimes squaring both sides of an equation can introduce extraneous solutions that don't actually satisfy the original equation.

6. Exponent rules: When simplifying expressions with exponents, remember the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

7. No negative square roots: In the real number system, the square root of a number cannot be negative. If you arrive at a solution that implies a negative square root, it means there is no solution in the real numbers.

8. Squaring a negative number: When you square a negative number, the result is positive. For example, $(-12)^2 = 144$.

9. Extraneous solutions: When you square both sides of an equation, you may introduce solutions that weren't there originally. Always check to make sure your solutions are valid for the original equation.