Equations with surds

Solve the following radical equation and check the validity of the solution obtained:

$\sqrt{3x - 2} = 4$

Determine the value of $x$ satisfying the following irrational equation:

$\sqrt{2x + 7} - x = 2$

Solve the following radical equation and explain why one of the potential algebraic solutions obtained after squaring is invalid:

$x - \sqrt{x + 1} = 5$

Aoife is solving the radical equation $\sqrt{3x + 1} = x - 1$ and presents the following steps on the board:

1) Squares both sides: $(\sqrt{3x + 1})^2 = (x - 1)^2 \implies 3x + 1 = x^2 - 2x + 1$.

2) Groups terms: $x^2 - 5x = 0 \implies x(x - 5) = 0$.

3) Concludes: "The solutions to the equation are $x = 0$ and $x = 5$".

1. a) Point out the conceptual error Aoife made in her conclusion and explain why the operation of squaring whole sides can introduce values that do not solve the original equation.
2. b) Write out the complete and correct resolution, carrying out a rigorous check of each candidate in the original equation to determine the final set of valid solutions.

Consider the equation with a radical $\sqrt{5x - 4} = x - 2$:

1. a) Without squaring, deduce a lower bound for the unknown taking into account that the principal square root takes non-negative values and that, therefore, $x - 2 \ge 0$ must hold. From that bound, estimate which of the integers $x = 1$, $x = 4$, or $x = 8$ can be the real solution by mentally evaluating both sides.
2. b) Solve the equation algebraically by isolating the radical and squaring both sides.
3. c) Check the obtained values in the original equation and explain how your previous estimate allowed you to discard beforehand the extraneous solution generated by the algebraic method.

Solve the following equation containing a higher-order root:

$\sqrt[3]{5x - 4} = -1$

Find the values of $x$ satisfying the following equation with fractions inside the radicand:

$\sqrt{\frac{4x - 1}{3}} = 5$

Find the solution to the equation by equating both radicals and check the result:

$\sqrt{x + 5} = \sqrt{3x - 1}$

Solve the following irrational equation containing two radicals:

$\sqrt{x + 4} - \sqrt{x - 1} = 1$

Find the real solutions to the following irrational equation by properly isolating one of the radicals before squaring:

$\sqrt{2x - 1} + \sqrt{x + 4} = 6$

Find the real solutions to the following equation containing a nested radical (a radical inside another):

$\sqrt{x + \sqrt{x + 3}} = 3$

Solve the following irrational equation. It will be easier if you use a suitable substitution, for example $t = \sqrt{x^2 - 2}$:

$x^2 - 3\sqrt{x^2 - 2} = 6$

Solve the following irrational equation with roots of different indices. It will be very helpful to perform the substitution $t = \sqrt[6]{x}$:

$\sqrt{x} - 7\sqrt[3]{x} + 12\sqrt[6]{x} = 0$

Solve the following irrational equation containing three radicals. Take special care when collecting terms after squaring for the first time:

$\sqrt{x + 2} + \sqrt{x - 3} = \sqrt{3x + 4}$

Solve the following equation, which combines the properties of absolute values with irrational expressions. Carefully consider the cases for the absolute value:

$|\sqrt{3x + 1} - 4| = 2$

Analyse and solve the following non-linear system of equations involving radicals. You can exploit the algebraic structure of the second equation (difference of two squares) to simplify the process:

$\begin{cases} \sqrt{x} + \sqrt{y} = 5 \\ x - y = 5 \end{cases}$

Prove algebraically and with reasoning why the irrational equation $\sqrt{x - 4} + \sqrt{2 - x} = 5$ has no solutions in the set of real numbers. Base your proof on the analysis of the domain of definition of the radicals involved, without needing to square both sides.

The time $t$ (in seconds) taken for an object to fall freely from a height $h$ (in metres) is given by the physics formula $t = \sqrt{\frac{2h}{9.8}}$. Calculate the height from which a stone was dropped if it took exactly $3$ seconds to reach the ground.

The base radius $r$ of a cylinder of volume $V$ and height $h$ is calculated using the expression $r = \sqrt{\frac{V}{\pi h}}$. If a cylindrical container of height $15\text{ cm}$ has a volume of $471\text{ cm}^3$, set up the corresponding equation and find its base radius (use the approximation $\pi \approx 3.14$).

The production cost $C$ (in thousands of euros) of manufacturing $x$ units of a high-precision electronic component follows the mathematical model $C(x) = 12 + \sqrt{5x + 4}$. If the exact budget allocated for the production of this batch is $25$ thousand euros, how many units can be manufactured?

The period $T$ of oscillation of a simple pendulum (in seconds) is related to its length $L$ (in metres) by the physical expression $T = 2\pi \sqrt{\frac{L}{g}}$. Taking the acceleration due to gravity as $g \approx \pi^2\text{ m/s}^2$, set up the equation and find the length of a pendulum whose period of oscillation is exactly $2.5$ seconds.

Calculate the $x$-coordinate where a point $P(x, 4)$ must be placed relative to point $A(-2, 1)$ so that the Euclidean distance between them is exactly $5$ units. Set up the corresponding radical equation using the distance formula and solve it.

In an architectural project, an access ramp is designed resting against a vertical wall $12\text{ m}$ high. The length of the ramp is $\sqrt{x^2 + 144}$ and the horizontal distance from the start of the ramp to the base of the wall is $x$. If, due to safety and gradient regulations, the ramp must measure exactly $4\text{ m}$ more than the horizontal distance $x$, set up and solve the irrational equation to find $x$.

Special relativity establishes that the relativistic mass $m$ of a moving object is given by the equation $m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass, $v$ is the velocity of the object and $c$ is the speed of light in a vacuum. Rearrange algebraically to find the velocity $v$ in terms of $m$, $m_0$ and $c$. Subsequently, calculate at what fraction of the speed of light $c$ a spacecraft must travel for its relativistic mass to be exactly double its rest mass.