Radical Equations
A rock is dropped from a bridge and splashes into the river exactly 3 seconds later. The fall time is $t=\sqrt{h/16}$, where $h$ is the height in feet. How high is the bridge? The unknown $h$ is stuck under a square root, and to get it out you have to undo the root. Equations like this are called radical equations. Some problems call them irrational equations.
Isolate and Square
Squaring undoes a square root: $(\sqrt{A})^2=A$ whenever $A\ge 0$. So the plan is to get the radical alone, then square.
Solve the following radical equation and check the validity of the solution obtained: $\sqrt{3x - 2} = 4$
Solve the following radical equation and check the validity of the solution obtained:
$\sqrt{3x - 2} = 4$
Extraneous Solutions
Squaring can add solutions that don't belong. For example, $3^2=9$ and $(-3)^2=9$, so squaring can't tell $3$ from $-3$. The equation $a=b$ implies $a^2=b^2$, but $a^2=b^2$ does not imply $a=b$.
The graph shows why. The curve $y=\sqrt{x+6}$ meets the line $y=x$ only once. Squaring also brings in the lower half $y=-\sqrt{x+6}$, and that half meets the line at $x=-2$.
Determine the value of $x$ that satisfies the following irrational equation: $\sqrt{2x + 7} - x = 2$
Determine the value of $x$ that satisfies the following irrational equation:
$\sqrt{2x + 7} - x = 2$
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 radical equation and explain why one of the possible algebraic solutions obtained after squaring is not valid:
$x - \sqrt{x + 1} = 5$
Other Indices and Fractions in the Radicand
Fourth roots behave like square roots. For example, $\sqrt[4]{x+1}=2$ gives $x+1=16$, so $x=15$. And $\sqrt[4]{x+1}=-2$ has no solution, because a fourth root is never negative.
Solve the following equation containing a higher-index root: $\sqrt[3]{5x - 4} = -1$
Solve the following equation containing a higher-index root:
$\sqrt[3]{5x - 4} = -1$
Find the values of $x$ that satisfy the following equality with fractions in the radicand: $\sqrt{\frac{4x - 1}{3}} = 5$
Find the values of $x$ that satisfy the following equality with fractions in the radicand:
$\sqrt{\frac{4x - 1}{3}} = 5$
Two Radicals
The graph of the left side, $y=\sqrt{x+5}+\sqrt{x}$, crosses the line $y=5$ exactly at $x=4$.
Find the solution to the equation by equating both radicals and verify the result: $\sqrt{x + 5} = \sqrt{3x - 1}$
Find the solution to the equation by equating both radicals and verify the result:
$\sqrt{x + 5} = \sqrt{3x - 1}$
Solve the following irrational equation containing two radicals: $\sqrt{x + 4} - \sqrt{x - 1} = 1$
Solve the following irrational equation containing two radicals:
$\sqrt{x + 4} - \sqrt{x - 1} = 1$
Squaring Twice
When one squaring leaves a radical behind, isolate that radical and square again. Extraneous solutions can sneak in at either squaring, so always check at the end in the original equation.
Find the real solutions of the following irrational equation by appropriately isolating one of the radicals before squaring: $\sqrt{2x - 1} + \sqrt{x + 4} = 6$
Find the real solutions of the following irrational equation by appropriately isolating one of the radicals before squaring:
$\sqrt{2x - 1} + \sqrt{x + 4} = 6$
Solve the following irrational equation containing three radicals. Take special care when grouping terms after the first squaring:…
Solve the following irrational equation containing three radicals. Take special care when grouping terms after the first squaring:
$\sqrt{x + 2} + \sqrt{x - 3} = \sqrt{3x + 4}$
Nested Radicals and Substitution
Some equations are really quadratics "in disguise". If the same radical expression appears twice, substitute $t$ for it.
Sometimes the radicand itself appears outside the radical. Then you must rewrite that outside part in terms of $t$ too. If $t=\sqrt{x^2-2}$, square both sides of this definition to get $t^2=x^2-2$, so $x^2=t^2+2$.
When the equation has roots of different indices, choose $t$ as a root whose index is the LCM of all the indices, so each radical becomes a power of $t$.
Find the real solutions of the following equation containing a nested radical (a radical inside another): $\sqrt{x + \sqrt{x + 3}} = 3$
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 $t = \sqrt{x^2 - 2}$:…
Solve the following irrational equation. You will find it easier if you apply an appropriate change of variable, for example $t = \sqrt{x^2 - 2}$:
$x^2 - 3\sqrt{x^2 - 2} = 6$
Solve the following irrational equation with roots of different indices. You will find it very helpful to make the change of variable $t = \sqrt[6]{x}$:…
Solve the following irrational equation with roots of different indices. You will find it very helpful to make the change of variable $t = \sqrt[6]{x}$:
$\sqrt{x} - 7\sqrt[3]{x} + 12\sqrt[6]{x} = 0$
Radicals Inside Absolute Value
Some problems call this a modulus equation. In US classes it's an absolute value equation. If $|y|=c$ with $c\ge0$, then $y=c$ or $y=-c$, because both numbers are $c$ units from zero.
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 equation that combines the properties of moduli with irrational expressions. Carefully consider the cases of the modulus:
$|\sqrt{3x + 1} - 4| = 2$
Systems with Radicals
A system with radicals is solved like any other system: use substitution or elimination, and use algebra to make the radicals disappear. Always remember that every radicand must be $\ge 0$.
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…
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:
$\begin{cases} \sqrt{x} + \sqrt{y} = 5 \\ x - y = 5 \end{cases}$
Domain, No Solution, and Parameters
Sometimes you can decide a lot before squaring anything. For an even index, the radicand must be $\ge0$. That condition is the domain of the equation.
Now suppose the equation has a parameter $k$. After squaring you get a quadratic whose roots depend on $k$. The discriminant tells you how many roots there are, and the sign condition on the isolated radical tells you which ones survive.
Move the slider and watch the curve $y=\sqrt{2x+k}$ meet the line $y=x$ in zero, one, or two points.
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…
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.
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…
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.
Applications: Formulas and Models
Many real formulas have a square root: horizon distance, fall time, pendulum period, cost models. To use them "backwards," isolate the radical, square, and solve. In real problems, discard values that don't make sense (negative length, time, or cost).
The graph above models a cost: $C(x)=8+\sqrt{3x+1}$ thousand dollars for $x$ units. Setting $C=15$ means finding where the curve meets the horizontal line $y=15$.
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…
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.
The production cost $C$ (in thousands of dollars) of manufacturing $x$ units of a high-precision electronic component follows the mathematical model…
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?
The period of oscillation $T$ of a simple pendulum (in seconds) is related to its length $L$ (in feet) by the physical expression $T = 2\pi \sqrt{\frac{L}{g}}$.…
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.
Applications: Distance and Geometry
The distance formula and the Pythagorean Theorem both produce square roots, so geometry problems often lead to radical equations.
In the next example, a ramp rests on a wall. The $x^2$ terms cancel after squaring, so you end up with a linear equation.
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…
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.
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…
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 $x$.
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 $P$…
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.
Solving a Formula for a Variable
A literal equation has several letters, and you solve for one of them. The method is the same: isolate the radical, raise to the index, then isolate the variable you want.
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…
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 $\pi \approx 3,14$).
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…
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.
Back to the start: the rock satisfies $3=\sqrt{h/16}$. Square both sides to get $9=h/16$, so $h=144$ ft. Check: $\sqrt{144/16}=\sqrt{9}=3$ ✓. The bridge is 144 feet high.