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 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 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) 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 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.

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

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

Find the solutions to the following equation, making sure to check the results obtained to rule out 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 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 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 $B$.

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 pesos as a function of $x$ units manufactured is modeled by the function $C(x) = 5\,000 + 300\sqrt{x - 50}$. If on a given day the production cost was exactly $17\,000$ pesos, how many units were manufactured?

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

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

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?

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 to the equation with nested radicals:

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

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

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.

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

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

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.

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 plant $A$ located on one side of a straight river $4$ km wide to a town $B$ located on the opposite side, $10$ km downstream. The cost of laying the cable underwater is $100$ million pesos per km, and on land it is $60$ 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 $C(x) = 100\sqrt{x^2 + 16} + 60(10 - x)$. Find the value of $x$ that makes the total cost exactly $920$ million pesos by solving the corresponding radical equation.