Exercises: Equations with radicals

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

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

Determine the value of $x$ that satisfies the following irrational equation:

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

Solve the following equation containing a higher-index root:

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

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

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

The time $t$ (in seconds) it takes for an object to fall freely from a height $h$ (in feet) is given by the physical formula $t = \sqrt{\frac{2h}{9,8}}$. Calculate from what height a stone was dropped if it took exactly $3$ seconds to reach the ground.

Find the values of $x$ that satisfy the following equality with fractions in the radicand:

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

The radius $r$ of the base 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 the radius of its base (use the approximation $\pi \approx 3,14$).

Solve the following radical equation and explain why one of the possible algebraic solutions obtained after squaring is not valid:

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

Solve the following irrational equation containing two radicals:

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

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

The production cost $C$ (in thousands of dollars) 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 dollars, how many units can be manufactured?

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

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

The period of oscillation $T$ of a simple pendulum (in seconds) is related to its length $L$ (in feet) by the physical expression $T = 2\pi \sqrt{\frac{L}{g}}$. Considering the acceleration due to gravity as $g \approx \pi^2\text{ ft/s}^2$, set up the equation and find the length of a pendulum whose period of oscillation is exactly $2,5$ seconds.

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

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

Solve the following irrational equation. You will find it easier if you apply an appropriate change of variable, for example $t = \sqrt{x^2 - 2}$:

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

In an architecture project, an access ramp is designed resting on a vertical wall of height $12\text{ m}$. 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 $x$.

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

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

A fiber optic cable must connect two communication towers $A$ and $B$ of height $15\text{ ft}$ and $20\text{ ft}$ respectively, passing through an anchor point $P$ located on the ground between the two towers. The horizontal distance between the towers is $35\text{ ft}$. If point $P$ is located $x$ feet from tower $A$, the total length of the cable is given by $L = \sqrt{x^2 + 225} + \sqrt{(35 - x)^2 + 400}$. If exactly $50\text{ ft}$ of cable is available and it is to be used in its entirety without cuts, determine at what distance $x$ the anchor $P$ must be placed.

Determine the numerical value of the parameter $k$ so that the irrational equation $\sqrt{x + k} = x - 1$ has a single valid real solution of multiplicity one. Justify your answer by analyzing the discriminant of the resulting quadratic equation and the validity conditions of the radical.

Solve the following equation that combines the properties of moduli with irrational expressions. Carefully consider the cases of the modulus:

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

Solve the following irrational equation with roots of different indices. You will find it very helpful to make the change of variable $t = \sqrt[6]{x}$:

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

The theory of special relativity states 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. Algebraically solve for the velocity $v$ in terms of $m$, $m_0$ and $c$. Subsequently, calculate at what fraction of the speed of light $c$ a spaceship must travel so that its relativistic mass is exactly double its rest mass.

Discuss and solve the following non-linear system of equations involving radicals. You can take advantage of 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.