Exercises: Irrational equations

Solve the following irrational equation and check the validity of its solution:

$\sqrt{4x + 9} = 5$

Find the value of $x$ that satisfies the following equation with a cube root:

$\sqrt[3]{3x - 5} = 2$

Determine the possible solutions of the quadratic equation under the radical:

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

Solve the irrational equation by isolating the radical and squaring. Pay special attention to possible false solutions:

$\sqrt{2x + 15} = x$

Find the solution by equating the radicands of the following equation:

$\sqrt{3x - 2} = \sqrt{x + 8}$

Solve for the unknown $x$ in the irrational equation containing a fraction in its radicand:

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

In kinematics, the time $t$ (in seconds) it takes for an object to fall freely from a height $h$ (in feet) is given by the formula $t = \sqrt{\frac{2h}{9.8}}$. Calculate from what height a stone was dropped if it took exactly $4$ seconds to reach the ground.

The diagonal $d$ of a rectangle can be calculated from its sides $b$ and $h$ using the Pythagorean theorem: $d = \sqrt{b^2 + h^2}$. If we know that the diagonal measures $13$ in and one of its sides measures $5$ in, set up the corresponding irrational equation and calculate the length of the other side.

Solve the equation with two radicals by appropriately isolating them on different sides before squaring:

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

Find the solutions of the following equation, checking the results obtained to discard extraneous solutions:

$x - \sqrt{2x - 1} = 2$

The period $T$ of a simple pendulum (the time it takes to complete one full oscillation) is calculated using the formula $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the string in feet and $g = 9.8 \text{ ft/s}^2$ is the acceleration due to gravity. If a pendulum clock in a science exhibition has an exact period of $2.5$ seconds, find the length of its pendulum. Use the approximation $\pi \approx 3.14$.

The distance between points $A(2, -1)$ and $B(x, 3)$ on the Cartesian plane is exactly $5$ units. Set up the corresponding irrational equation for the distance between two points and calculate the possible coordinates $x$ of point $B$.

Solve the following equation involving the difference of two square roots. Isolate one of the roots before squaring:

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

In an electronic components factory, the daily production cost $C(x)$ in dollars as a function of $x$ manufactured units is modeled by the function $C(x) = 250 + 15\sqrt{x - 50}$. If on a given day the production cost was exactly $\$850$, how many units were manufactured?

Solve the following irrational equation by making an appropriate change of variable (for example, $t = \sqrt{x}$):

$x - 7\sqrt{x} + 10 = 0$

The distance $d$ in miles to the horizon for an observer located at a height $h$ in feet above sea level is approximated in a simplified way by the formula $d = 3.57\sqrt{h}$. If a lookout in the crow's nest of a pirate ship sights land at a distance of $15$ miles, at what height are they approximately?

Solve the following irrational equation containing three radicals. Make sure to check the solutions obtained at the end of the process:

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

Find the real solutions of the equation with nested radicals:

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

Discuss with reasoning for which values of the real parameter $k > 0$, the equation $\sqrt{x + k} = x$ has exactly one valid real solution.

An engineer designs a loading ramp for a logistics warehouse. The horizontal base measures $x$ feet and the height measures $2$ feet less than the base. If the length of the ramp (the hypotenuse) is given by the expression $\sqrt{2x^2 - 4x + 4}$ and it is required to measure exactly $\sqrt{34}$ feet, set up the irrational equation and determine the real dimensions of the ramp.

Solve the following irrational equation. You can rationalize the denominator beforehand, or cross-multiply and group terms intelligently to facilitate the calculation:

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

According to Einstein's Theory of Special Relativity, the Lorentz factor $\gamma$ determines time dilation and is calculated as $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is the speed of the spaceship and $c$ is the speed of light in a vacuum. If on a space mission it is observed that time passes at $50\%$ of the Earth's rate (that is, $\gamma = 2$), solve the irrational equation to find at what fraction of the speed of light the spaceship is traveling.

Solve the following irrational equation by strategically isolating the radicals and verifying the solutions at the end:

$\sqrt{2x + 5} - \sqrt{x - 1} = \sqrt{x + 2}$

A power line must connect a power station $A$ located on one side of a straight river $4$ miles wide, with a city $B$ located on the opposite side, $10$ miles downstream. The cost of laying the cable underwater is $\$5$ million per mile, and over land it is $\$3$ million per mile. If the cable crosses the river in a straight line to a point $P$ located $x$ miles downstream and then goes over land to $B$, the total cost in millions of dollars is $C(x) = 5\sqrt{x^2 + 16} + 3(10 - x)$. Find the value of $x$ that makes the total cost exactly $\$46$ million by solving the corresponding irrational equation.