Solving Radical Equations · Mathematics Integrated Math III

Isolate, raise to a power, check

Some problems call these irrational equations. It is the same thing.

The graph shows the same story: the curve $y=\sqrt{x+6}$ and the line $y=x$ cross only once.

Practice★☆☆

Solve the following irrational equation and check the validity of its solution: $\sqrt{4x + 9} = 5$

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

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

Solve

Practice★☆☆

Solve the irrational equation by isolating the radical and squaring. Pay special attention to possible false solutions: $\sqrt{2x + 15} = x$

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

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

Solve

Practice★★☆

Find the solutions of the following equation, checking the results obtained to discard extraneous solutions: $x - \sqrt{2x - 1} = 2$

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

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

Solve

Other radicands and other indexes

Practice★☆☆

Find the value of $x$ that satisfies the following equation with a cube root: $\sqrt[3]{3x - 5} = 2$

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

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

Solve

Practice★☆☆

Determine the possible solutions of the quadratic equation under the radical: $\sqrt{x^2 - 3x + 6} = 2$

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

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

Solve

Practice★☆☆

Solve for the unknown $x$ in the irrational equation containing a fraction in its radicand: $\sqrt{\frac{5x - 1}{3}} = 3$

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

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

Solve

Two radicals

Practice★☆☆

Find the solution by equating the radicands of the following equation: $\sqrt{3x - 2} = \sqrt{x + 8}$

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

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

Solve

Practice★★☆

Solve the equation with two radicals by appropriately isolating them on different sides before squaring: $\sqrt{x + 5} + \sqrt{x} = 5$

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

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

Solve

Practice★★☆

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

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

Solve

Harder cases: three radicals, nested radicals, fractions

These use the same ideas. Isolate, square, repeat, and check at the end. Each squaring is another chance to introduce a false solution, and checking is the only real guarantee.

Practice★★★

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

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

Practice★★★

Find the real solutions of the equation with nested radicals: $\sqrt{x + \sqrt{x - 2}} = 2$

Find the real solutions of the equation with nested radicals:

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

Solve

Practice★★★

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

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

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

Solve

Substitution: quadratic in form

Practice★★☆

Solve the following irrational equation by making an appropriate change of variable (for example, $t = \sqrt{x}$): $x - 7\sqrt{x} + 10 = 0$

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

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

Solve

Domain and parameters: reasoning without solving

Every expression under an even-index radical must be $\ge0$. Sometimes that alone shows an equation is impossible, with no squaring at all.

When an equation has a parameter, the question is usually "for which values of $k$ is there a solution, and how many?" The plan:

Square and solve the resulting quadratic. The discriminant tells you how many candidates exist.
Apply the sign condition from the original equation (the side without a radical must be $\ge0$) to each candidate.
Count the survivors for each range of $k$.

Practice★★★

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

Solve

Formulas and models

Many science formulas contain a square root. To find a quantity inside the radical, isolate the radical (divide by anything multiplying it) and square.

Practice★☆☆

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…

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.

Solve

Practice★★☆

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…

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

Practice★★★

According to Einstein's Theory of Special Relativity, the Lorentz factor $\gamma$ determines time dilation and is calculated as…

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

Geometry, distance, and cost

The Pythagorean Theorem and the distance formula both produce a square root, so they lead directly to radical equations.

Cost models with a radical

Cost problems usually combine a distance that comes from the Pythagorean Theorem (a radical) with a distance that is simply a difference (no radical). Each piece of cable costs price per mile $\times$ miles, and the total cost is the sum.

Cable cost across a river

A power station $A$ is on one bank of a river that is $3$ miles wide. A city $B$ is on the opposite bank, $8$ miles downstream from the point directly across from $A$. Cable costs $\$4$ million per mile under water and $\$2$ million per mile on land. The cable goes straight through the water to a point $P$ that is $x$ miles downstream, then over land to $B$. For what $x$ does the total cost equal $\$28$ million?

Set up the cost. Under water the cable is the hypotenuse of a right triangle with legs $3$ and $x$, so its length is $\sqrt{x^2+9}$ miles. On land it is $8-x$ miles. Total cost in millions of dollars:

$$C(x)=4\sqrt{x^2+9}+2(8-x)$$

Set $C(x)=28$: $4\sqrt{x^2+9}+2(8-x)=28$.

Isolate the radical. Move the land term to the right: $4\sqrt{x^2+9}=28-16+2x=12+2x$. Divide by $2$: $2\sqrt{x^2+9}=6+x$. Sign condition: $6+x\ge0$.

Square and solve. $4(x^2+9)=(6+x)^2$, so $4x^2+36=36+12x+x^2$. Then $3x^2-12x=0$, so $3x(x-4)=0$ and $x=0$ or $x=4$.

Check in the original equation and in the context. The landing point must be between $0$ and $8$ miles, and both values are. $x=0$: $4\cdot3+2\cdot8=12+16=28$ ✓. $x=4$: $4\cdot5+2\cdot4=20+8=28$ ✓. Both are valid. The same total cost happens at two landing points, because the cost dips below $\$28$ million somewhere in between.

This is exactly the pattern for any cost model like $C(x)=a\sqrt{x^2+b}+c(d-x)$ set equal to a number: set up the radical from the right triangle, move every term without the radical to the other side, isolate the radical, square, and check in the original equation (and in the context).

Practice★☆☆

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…

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

Practice★★☆

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…

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

Practice★★★

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…

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.

Solve

Back to the start: $x=-2$ sneaked in because squaring $\sqrt{x+6}=x$ also solves $\sqrt{x+6}=-x$, and $x=-2$ is a solution of that one. Only the check in the original equation tells them apart.

Summary

Isolate the radical, raise both sides to the power of the index, solve, and always check every candidate in the original equation.
Squaring can add extraneous solutions. If the isolated radical equals an expression that comes out negative, reject the candidate at once.
With two radicals, isolate one, square, isolate the one that is left, and square again. Expand binomials fully, including the middle term.
If $x$ appears only as $x$ and $\sqrt{x}$, let $t=\sqrt{x}$ ($t\ge0$), solve the quadratic in $t$, and go back with $x=t^2$.
The domain (each even-index radicand $\ge0$) can show there is no solution without solving. With a parameter, use the discriminant and then the sign condition.
In applications, build the radical from the Pythagorean Theorem or the distance formula, and also reject solutions that don't fit the context, such as negative lengths.

Let's practice! 8 medium exercises in this section

★★☆ Solve the equation with two radicals by appropriately isolating them on different sides…★★☆ Find the solutions of the following equation, checking the results obtained to discard…★★☆ The period $T$ of a simple pendulum (the time it takes to complete one full oscillation)…

Let's practice! 8 hard exercises in this section

★★★ Solve the following irrational equation containing three radicals. Make sure to check the…★★★ Find the real solutions of the equation with nested radicals:…★★★ Discuss with reasoning for which values of the real parameter $k > 0$, the equation…