This question requires rationalizing the denominator of the fraction 12/√3 to eliminate the radical from the denominator. To rationalize, we multiply both numerator and denominator by √3: (12/√3) × (√3/√3) = 12√3/3. Finally, we simplify the fraction by dividing: 12√3/3 = 4√3.

This question asks us to solve the radical equation √(x + 1) = 4 by isolating the variable under the radical. To eliminate the square root, we square both sides: (√(x + 1))² = 4², which gives us x + 1 = 16. Solving for x: x = 16 - 1 = 15.

This question asks us to find an equivalent expression for √18 + √8 by simplifying each radical separately and then combining like terms. First, simplify √18 = 3√2 and √8 = 2√2. Now we can add the like terms: 3√2 + 2√2 = 5√2.

This question asks us to simplify the radical expression √72 by factoring out perfect squares. First, we find the prime factorization: 72 = 36 × 2. Since 36 is a perfect square, we can extract it from under the radical: √72 = √(36 × 2) = √36 × √2 = 6√2.

1. Square both sides: $x + 14 = (x + 2)^2$.

2. Expand: $x + 14 = x^2 + 4x + 4$.

3. Set to zero: $x^2 + 3x - 10 = 0$.

4. Factor: $(x + 5)(x - 2) = 0$.

5. Potential solutions: $x = -5, x = 2$.

6. Check for extraneous solutions:

   - If $x = -5$: $\sqrt{9} = -3$ (False, principal root must be positive).

   - If $x = 2$: $\sqrt{16} = 4$ (True).

Only $x=2$ is valid.

SAT Strategy: Backsolve

Plug the answer choices into the equation.

A) $\sqrt{-5+14} = -5+2 \rightarrow 3 = -3$ (False).

B) $\sqrt{2+14} = 2+2 \rightarrow 4 = 4$ (True).

Look for perfect cubes within each term. This will allow us to factor out of the radical.

Simplify.

Even if you're unsure of where to start on this problem, you should have a head start. The problem is testing absolute values, and you should know that the result of any absolute value is always nonnegative ≥0. So the answer choices that include an absolute value equaling a negative number must be incorrect: that just cannot be possible.

To test the remaining choices, consider that the numbers that are exactly eight units away from -2 are -2+8 = 6, and -2-8 = -10.  When you plug these numbers in for x in the answer choices, only one is valid:

$|x+2|=8$

For $x=6$, $|6+2|=|8|=8$

For $x=-10$, $|-10+2|=|-8|=8$

Therefore this absolute value satisfies the given situation and is correct.

Absolute value equations split into two cases:

1. $2x - 7 = 13 \implies 2x = 20 \implies x = 10$.

2. $2x - 7 = -13 \implies 2x = -6 \implies x = -3$.

3. Sum of solutions: $10 + (-3) = 7$.

Now lets plug these values into our original equation.

For

So this isn't an extraneous solution.

For

Since , is an extraneous solution.

First step is to set it equal to zero

Now factor out an

Set up the equations, and solve for .