Radical equations and inequalities

9000024802

Level: 
A
Consider the equation \[ \sqrt{x^{2 } - 2x + 1} = x + 2 \] and the equation which arises from this equation by squaring both sides of the equation, i.e. the equation \[ \left (\sqrt{x^{2 } - 2x + 1}\right )^{2} = (x + 2)^{2}. \] Identify a true statement.
Both equations are equivalent only if \(x\geq - 2\).
Both equations are equivalent.
Both equations are equivalent only if \(x\leq - 2\).
None of the above.

9000024803

Level: 
A
Removing radical in an equation by squaring both sides may enrich the set of solutions of this equation and checking the solutions of the new equation in the original equation may be necessary. Identify a correct conclusion in the particular case of the following equation. \[ -\sqrt{x^{2 } - 2x + 1} = x \]
If we look for the solution in the set \(\mathbb{R}^{-}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
If we look for the solution in the set \(\mathbb{R}^{+}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
If we look for the solution in the set \(\mathbb{R}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
None of the above.

9000023710

Level: 
A
Identify a true statement which concerns the following pair of equations. \[ \begin{aligned} \sqrt{ 2x + 17} & = 3 &\text{(1)} \\ \sqrt{8 - 4x} & = 4 &\text{(2)} \end{aligned} \]
The product of the solutions of (1) and (2) is \(8\).
The sum of the solutions of (1) and (2) is \(- 2\).
The quotient of the solution of (1) divided by the solution of (2) is \(- 2\).
The quotient of the solution of (2) divided by the solution of (1) is \(- 0.5\).

9000023803

Level: 
A
In the following list identify a true statement referring to the solution of the following equation. \[ \sqrt{x + 3} = 3 + x \]
The difference of the bigger and smaller solutions is \(1\).
The difference of the bigger and smaller solutions is \(- 1\).
The difference of the smaller and the bigger solutions is \(1\).
The difference of the smaller and twice the bigger solutions is \(- 1\).

9000023804

Level: 
A
Identify a true statement which concerns to the following equation. \[ \sqrt{x + 3} = x - 3 \]
The solution is in the interval \((5;8)\).
The solution is in the interval \([ - 2;2] \).
The solution is in the interval \([ - 3;1)\).
The solution is in the interval \([ 3;5)\).

9000022305

Level: 
A
Find the domain of the following expression. \[ \sqrt{-x^{2 } + 16x - 63} \]
\(\left [ 7;9\right ] \)
\(\left (-\infty ;7\right )\cup \left (9;\infty \right )\)
\(\left (-\infty ;-7\right ] \cup \left [ 9;\infty \right )\)
\(\left (7;9\right )\)
\(\left [ -7;9\right ] \)