Irrational Equations

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

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

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

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

Find the solutions to the following equation, making sure to check the results obtained to rule out extraneous solutions:

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

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

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

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

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

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

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

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

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

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

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

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

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

Solve the following irrational equation by making a suitable substitution (for example, $t = \sqrt{x}$):

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

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

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

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

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

Find the real solutions to the equation with nested radicals:

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

Solve the following irrational equation. You can rationalise the denominator first, or cross-multiply and collect terms cleverly to facilitate the calculation:

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

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

In kinematics, the time $t$ (in seconds) taken for an object to fall freely from a height $h$ (in meters) 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 hit the ground.

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 meters and $g = 9.8 \text{ m/s}^2$ is the acceleration due to gravity. If a pendulum clock at 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 $d$ in kilometers to the horizon for an observer situated at a height $h$ in meters above sea level is approximated in simplified form by the formula $d = 3.57\sqrt{h}$. If a lookout in the crow's nest of a pirate ship spots land at a distance of $15$ km, at approximately what height is he situated?

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

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

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

In an electronic components factory, the daily production cost $C(x)$ in pesos as a function of $x$ units manufactured is modeled by the function $C(x) = 15\,000 + 900\sqrt{x - 50}$. If on a given day the production cost was exactly $51\,000$ pesos, how many units were manufactured?

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 spacecraft and $c$ is the speed of light in a vacuum. If in a space mission time is observed to pass 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 spacecraft is travelling.

A power plant must connect to a town $B$ located on the opposite side of a straight river $4$ km wide, $10$ km downstream from the power station $A$. The cost of laying the cable underwater is $300$ million pesos per km, and on land it is $180$ million pesos per km. If the cable crosses the river in a straight line to a point $P$ located $x$ km downstream and then goes along land to $B$, the total cost in millions of pesos is $C(x) = 300\sqrt{x^2 + 16} + 180(10 - x)$. Find the value of $x$ that makes the total cost exactly $2\,760$ million pesos by solving the corresponding radical equation.