Algebraic Equations with Square Roots

Introduction

As Curzon prepares for the upcoming AMC 10/12, I’m trying to make an “exhaustive” search, or categorize techniques that are used in solving algebra problems involving square roots. The following problems are approximately organized by difficulty level within each section, starting with basic manipulation and progressing to advanced techniques.

Warmup

Please don’t square anything yet!

1. 2017 UKMT SMC Problem 7: The positive integer $k$ satisfies the equation $\sqrt{2}+\sqrt{8}+\sqrt{18}=\sqrt{k}$. What is the value of $k$? A) 28 B) 36 C) 72 D) 128 E) 288

2. 2019 BmMT Team Problem 18: What is the smallest positive integer $x$ such that there exists an integer $y$ with $\sqrt{x}+\sqrt{y}=\sqrt{1025}$?

3. 2021 BmMT Team Problem 18: Compute the number of positive integers $1<k<2021$ such that the equation $x+\sqrt{k x}=k x+\sqrt{x}$ has a positive rational solution for $x$.

Infinite Nested Radicals

These problems involve expressions of the form $\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}$ where the pattern continues infinitely. The key insight is that if we call the entire expression $x$, then $x = \sqrt{a+x}$, leading to the equation $x^2 = a+x$.

Some of these problems also involve finite nested radicals, but they telescope or simplify in some way. Of these, only the final problem is really challenging.

1. 2016 BmMT Team Problem 13: The following expression is an integer. Find this integer: $\frac{\sqrt{\,20 + 16 \sqrt{\frac{20 + 16 \sqrt{\frac{20 + 16 \cdots}{2}}}{2}}\,}}{2}$.

2. 2006 HMMT February General Part 1 Problem 4: Find $\frac{\sqrt{31+\sqrt{31+\sqrt{31+\ldots}}}}{\sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}}$

3. 2024 MAΘ Hustle Algebra 2 Problem 12: Solve for $x$: $x=\sqrt{132-\sqrt{132-\cdots}}$

4. 2003 MAΘ Theta Gemini Problem 25: If $\sqrt{20-\sqrt{20-\sqrt{20-\sqrt{\ldots}}}}=x$ then $x=$

5. 2013 MAΘ Mu Ciphering Problem 4: Evaluate: $\sqrt{10 + 3\sqrt{10 + 3\sqrt{10 + \cdots}}}$

6. 2023 MAΘ Theta Equations and Inequalities Problem 6: $x=\sqrt{4+3\sqrt{4+3\sqrt{4+\cdots}}}$ Find $x$.

7. 2011 Crux Mayhem Problem M452 Part (a): Suppose that $x=\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}$ for some real number $a>0$. Prove that $x^{2}-a=x$.

8. 2011 Crux Mayhem Problem M452 Part (b): Determine the integer equal to $\frac{\sqrt{30+\sqrt{30+\sqrt{30+\cdots}}}}{\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}}-\sqrt{42+\sqrt{42+\sqrt{42+\cdots}}}$

9. 2001 HMMT February Guts Problem 14: Find the exact value of $1 + \cfrac{1}{1 + \cfrac{2}{1 + \cfrac{1}{1 + \cfrac{2}{1 + \ddots}}}}$.

10. 2008 HMMT February Guts Problem 17: Solve the equation $\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}}}-\sqrt{x}=1$

11. 2012 Crux Mayhem Problem M487: Let $m$ be a positive integer. Find all real solutions to the equation

$$m+\sqrt{m+\sqrt{m+\cdots \sqrt{m+\sqrt{m+\sqrt{x}}}}}=x$$

in which the integer $m$ occurs $n$ times.

Basic Radical Isolation

These problems involve isolating radical terms and squaring to eliminate them. The key technique is:

• Isolate one radical term on one side of the equation
• Square both sides to eliminate that radical
• If there’s another radical, repeat the process
• Always check for extraneous solutions introduced by squaring

1. 2009 MAΘ Hustle Algebra Problem 15: Solve: $x=2\sqrt{x}+24$

Solution

Solution 1 (Isolate and Square).

Isolate the radical:

$$x - 24 = 2\sqrt{x}$$

Square both sides:

$$(x - 24)^2 = 4x$$

Expand:

$$x^2 - 48x + 576 = 4x$$

Simplify:

$$x^2 - 52x + 576 = 0$$

Solve the quadratic:

$$x = \frac{52 \pm \sqrt{52^2 - 4\cdot576}}{2} = \frac{52 \pm \sqrt{400}}{2} = \frac{52 \pm 20}{2}$$

So

$$x = \frac{72}{2} = 36 \quad\text{or}\quad x = \frac{32}{2} = 16$$

Check $x=16$:

$$16 \stackrel{?}{=} 2\sqrt{16} + 24 = 8 + 24 = 32$$

This fails, so $x=16$ is extraneous. The valid solution is

$$x = 36$$

Solution 2 (Substitution).

Let

$$y = \sqrt{x}$$

so

$$x = y^2$$

Substitute:

$$y^2 = 2y + 24$$

Rewrite:

$$y^2 - 2y - 24 = 0$$

Factor:

$$(y - 6)(y + 4) = 0$$

Then

$$y = 6 \quad\text{or}\quad y = -4$$

Because $y = \sqrt{x} \ge 0$, take $y=6$. Thus

$$x = y^2 = 36$$

Note. Both solutions square exactly once. The isolation method squares the original equation after isolating the radical, while the substitution method first rewrites the problem as an equation linear in $\sqrt{x}$ and then squares that simpler form. Always check for extraneous roots introduced by squaring.

2. 2008 MAΘ Hustle Algebra 2 Problem 25: Solve $\sqrt{x-4}+x=6$

Solution

Solution 1 (Isolate and Square).

Same as above: isolate the radical, square, and solve the resulting quadratic.

Solution 2 (Substitution).

The substitution is a little trickier here. Let

$$y = \sqrt{x-4}$$

Then

$$x = y^2 + 4$$

Substitute:

$$y + (y^2 + 4) = 6$$

Simplify:

$$y^2 + y + 4 = 6$$ $$y^2 + y - 2 = 0$$

Factor:

$$(y+2)(y-1) = 0$$

So

$$y = -2 \quad\text{or}\quad y = 1$$

Since $y = \sqrt{x-4} \ge 0$, take $y=1$. Then

$$x = y^2 + 4 = 1 + 4 = 5$$

3. 2018 MAΘ Hustle Algebra Problem 21: Find all solutions to $x=\sqrt{3x+4}$

4. 2022 MAΘ Hustle Algebra 2 Problem 15: Find the sum of the solutions to $x = \sqrt{2x + 35}$

5. 2009 MAΘ Theta Individual Problem 11: Solve: $\sqrt{x-3}-x=-3$

6. 2009 MAΘ Theta Radicals Problem 9: Solve: $\frac{1}{2+\sqrt{x}}+3\sqrt{x}=\frac{19}{8}$

7. 2018 MAΘ Alpha Systems of Equations Problem 29: Solve: $\sqrt{x+2}+\sqrt{x}=\sqrt[5]{32}$.

Solution

Solution 1 (Three quick algebra routes). Two radicals make this interesting since you will have to square twice. You can isolate either one, or square right away. One of these choices is easier than the others:

• Isolate $\sqrt{x+2}$: from $\sqrt{x+2}+\sqrt{x}=2$ get $\sqrt{x+2}=2-\sqrt{x}$. Square: $x+2=4-4\sqrt{x}+x \Rightarrow 4\sqrt{x}=2 \Rightarrow \sqrt{x}=\tfrac12 \Rightarrow x=\boxed{\tfrac14}$.

• Isolate $\sqrt{x}$: from $\sqrt{x}=2-\sqrt{x+2}$. Square: $x=4-4\sqrt{x+2}+x+2 \Rightarrow 4\sqrt{x+2}=6 \Rightarrow \sqrt{x+2}=\tfrac32 \Rightarrow x+2=\tfrac94 \Rightarrow x=\boxed{\tfrac14}$.

• Square immediately: $(\sqrt{x+2}+\sqrt{x})^2=4 \Rightarrow (x+2)+x+2\sqrt{x(x+2)}=4 \Rightarrow 2x+2+2\sqrt{x(x+2)}=4 \Rightarrow 2\sqrt{x(x+2)}=2-2x \Rightarrow \sqrt{x(x+2)}=1-x$. Square again: $x(x+2)=(1-x)^2=1-2x+x^2 \Rightarrow x^2+2x=1-2x+x^2 \Rightarrow 4x=1 \Rightarrow x=\tfrac14$. Check: $x=\tfrac14$ satisfies the original equation, so the solution is $\boxed{\tfrac14}$.

Solution 2 (Substitution). Let $y=\sqrt{x}$ and $z=\sqrt{x+2}$. Then $y+z=2$ and $z^2-y^2=2 \Rightarrow (z-y)(z+y)=2 \Rightarrow (z-y)\cdot2=2 \Rightarrow z-y=1$. Solve $y+z=2$, $z-y=1$: adding gives $2z=3 \Rightarrow z=\tfrac32$, then $y=\tfrac12$. Thus $x=\boxed{\tfrac14}$.

Which version felt easiest? In general it pays to isolate the more complicated radical before squaring. Here the two radicals are similar, so isolating either works cleanly while squaring immediately is slightly more messy.

7. 2019 BmMT Individual Problem 13: If $x$ is a real number such that $\sqrt{x}+\sqrt{10}=\sqrt{x+20}$, compute $x$.

8. 2015 MAΘ Hustle Algebra 2 Problem 23: Solve for $x$ if $\sqrt{11-x} = \sqrt{-5x} + 1$.

9. 2013 MAΘ Mu State Bowl Problem 1: Find $x$ as a common fraction: $4+\sqrt{10-x}=6+\sqrt{4-x}$

10. 2015 JHMT Algebra Problem 3: Find the unique $x>0$ such that $\sqrt{x}+\sqrt{x+\sqrt{x}}=1$.

11. 2009 MAΘ Alpha Ciphering Problem 4: Find the sum of all real $x$ such that $\sqrt{\frac{x+4}{x-1}}+\sqrt{\frac{x-1}{x+4}}=\frac{5}{2}$.

Solution

Solution 1 (Three algebra routes). Two radicals mean you will square twice. You can isolate either one, or square right away.

• Isolate $\sqrt{x+\sqrt{x}}$: from $\sqrt{x}+\sqrt{x+\sqrt{x}}=1$ get $\sqrt{x+\sqrt{x}}=1-\sqrt{x}$. Square: $x+\sqrt{x}=(1-\sqrt{x})^{2}=1-2\sqrt{x}+x \Rightarrow 3\sqrt{x}=1 \Rightarrow \sqrt{x}=\tfrac13 \Rightarrow x=\boxed{\tfrac19}$.

• Isolate $\sqrt{x}$: from $\sqrt{x}=1-\sqrt{x+\sqrt{x}}$. Square: $x=1-2\sqrt{x+\sqrt{x}}+x+\sqrt{x} \Rightarrow 2\sqrt{x+\sqrt{x}}=1+\sqrt{x}$. Subtract twice the original equation $2(\sqrt{x}+\sqrt{x+\sqrt{x}})=2$ to eliminate the complicated radical: $2\sqrt{x+\sqrt{x}}-(2\sqrt{x+\sqrt{x}})=1+\sqrt{x}-2\sqrt{x}=1-\sqrt{x}$ simplifies to $\sqrt{x}=3x$ after squaring, which gives $9x^{2}-x=0 \Rightarrow x\in\{0,\tfrac19\}$. Only $x=\tfrac19$ satisfies the original, so $x=\boxed{\tfrac19}$.

• Square immediately: $(\sqrt{x}+\sqrt{x+\sqrt{x}})^{2}=1 \Rightarrow 2x+\sqrt{x}+2\sqrt{x}\sqrt{x+\sqrt{x}}=1 \Rightarrow 2\sqrt{x}\sqrt{x+\sqrt{x}}=1-2x-\sqrt{x}$. Square: $4x(x+\sqrt{x})=(1-2x-\sqrt{x})^{2} \Rightarrow 2\sqrt{x}+3x-1=0 \Rightarrow \sqrt{x}=\tfrac{1-3x}{2}$. Square: $4x=(1-3x)^{2} \Rightarrow 9x^{2}-10x+1=0 \Rightarrow x\in\{1,\tfrac19\}$. Check the original; only $x=\tfrac19$ works, hence $x=\boxed{\tfrac19}$.

Solution 2 (One-variable substitution). Let $y=\sqrt{x}$. Then $y+\sqrt{y^{2}+y}=1 \Rightarrow \sqrt{y^{2}+y}=1-y$. Square: $y^{2}+y=(1-y)^{2}=1-2y+y^{2} \Rightarrow 3y=1 \Rightarrow y=\tfrac13 \Rightarrow x=y^{2}=\boxed{\tfrac19}$.

11. 2014 MMATHS Mathathon Problem 11: Compute $\sqrt{6-\sqrt{11}}-\sqrt{6+\sqrt{11}}$.

Solution

Square the expression:

$$\begin{aligned} (\sqrt{6-\sqrt{11}}-\sqrt{6+\sqrt{11}})^2 &= (6-\sqrt{11})+(6+\sqrt{11})-2\sqrt{(6-\sqrt{11})(6+\sqrt{11})} \\ &= 12 - 2\sqrt{36-11} \\ &= 12 - 2\sqrt{25} \\ &= 12 - 10 \\ &= 2 \end{aligned}$$

Since $\sqrt{6-\sqrt{11}} < \sqrt{6+\sqrt{11}}$, the difference is negative. Hence when we take the square root, the answer is the negative root:

$\sqrt{6-\sqrt{11}}-\sqrt{6+\sqrt{11}} = -\sqrt{2}$

26. 2016 MAΘ Theta Ciphering Problem 3: Solve for real $x$, where all expressions are real: $\sqrt{3x}+\sqrt{2x-1}=\frac{5}{\sqrt{2x-1}}$

27. 2009 MAΘ Theta Quadratics Problem 15: Find the sum of the value(s) of $x$ in $x^{2}+3x-\sqrt{x^{2}+3x}-6=0$.

28. 2003 MAΘ Alpha Ciphering Problem 9: Find $x$ if $\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}=x\sqrt{x}$.

29. 2023 BmMT Team Problem 15: Compute the product of all real solutions $x$ to the equation $x^{2}+20 x-23=2 \sqrt{x^{2}+20 x+1}$.

30. 2021 BmMT Individual Problem 16: The equation $\sqrt{x}+\sqrt{20-x}=\sqrt{20+20 x-x^{2}}$ has 4 distinct real solutions, $x_{1}, x_{2}, x_{3}$, and $x_{4}$. Compute $x_{1}+x_{2}+x_{3}+x_{4}$.

31. 2015 MMATHS Mathathon Problem 17: Find the sum of the distinct real roots of the equation $\sqrt[3]{x^{2}-2 x+1}+\sqrt[3]{x^{2}-x-6}=\sqrt[3]{2 x^{2}-3 x-5}$

32. 2017 Purple Comet High School Problem 20: Let $a$ be a solution to the equation $\sqrt{x^{2}+2}=\sqrt[3]{x^{3}+45}$. Evaluate the ratio of $\frac{2017}{a^{2}}$ to $a^{2}-15 a+2$.

Factoring/Completing the Square

Key techniques involve one of the following:

• Completing the square (e.g. $x^2+20x=(x+10)^2-100$),
• Factoring quadratics, cubics, or even quartics (often depressed),
• “Denesting” mixed radicals, e.g. $\sqrt{a+2\sqrt{b}}=\sqrt{m}+\sqrt{n}$ if $m+n=a$ and $mn=b$ (e.g. $\sqrt{5+2\sqrt{6}}=\sqrt{3}+\sqrt{2}$).

14. 2023 MMATHS Lightning Problem 78: What is the largest integer value of $x$ such that $\sqrt{x^{6}+2 x^{3}+1}-x^{3}>1$?
15. 2013 BMT Team Problem 4: Find the sum of all real numbers $x$ such that $x^{2}=5 x+6 \sqrt{x}-3$.
16. 2016 BmMT Team Problem 20: Find $x$ such that $\sqrt{c+\sqrt{c-x}}=x$ when $c=4$.
17. 2009 Crux Mayhem Problem M386: Determine all real numbers $x$ for which $\sqrt{2+4 x-2 x^{2}}+\sqrt{6+6 x-3 x^{2}}=x^{2}-2 x+6$
18. 2018 MAΘ Alpha Systems of Equations Problem 18: Find the sum of solutions to $\sqrt{x^{4}-6x^{2}+9}=2x$.
19. 2000 Crux Mayhem High School Problem H253: Find all real solutions to the equation $\sqrt{3 x^{2}-18 x+52}+\sqrt{2 x^{2}-12 x+162}=\sqrt{-x^{2}+6 x+280}$
20. 2000 Crux Mayhem High School Problem H261: Solve for $x$: $(\sqrt{7-\sqrt{48}})^{x}+(\sqrt{7+\sqrt{48}})^{x}=14$
21. 2021 BMT General Problem 17: Simplify $\sqrt[4]{17+12 \sqrt{2}}-\sqrt[4]{17-12 \sqrt{2}}$.
22. 2025 MAΘ Alpha School Bowl Problem 4B: Given $\sqrt{x+1-4\sqrt{x-3}}+\sqrt{x+6-6\sqrt{x-3}}=1$, how many integral solutions for $x$ exist?

Rationalizing the Denominator

22. 1980 ARML Team Problem 9: Find all values of $x$ which satisfy $\frac{6}{\sqrt{x-8}-9}+\frac{1}{\sqrt{x-8}-4}+\frac{7}{\sqrt{x-8}+4}+\frac{12}{\sqrt{x-8}+9}=0$
23. 2012 BmMT Ciphering Problem 40: Given that $x=\sqrt[3]{4}+\sqrt[3]{2}+\sqrt[3]{1}$, what is the value of $\frac{3}{x}+\frac{3}{x^{2}}+\frac{1}{x^{3}}$?

The Old x+1/x Trick

21. 2014 Purple Comet High School Problem 23: Suppose $x$ is a real number satisfying $x^{2}-990 x+1=(x+1) \sqrt{x}$. Find $\sqrt{x}+\frac{1}{\sqrt{x}}$.
22. 2012 Purple Comet High School Problem 24: There are positive integers $m$ and $n$ so that $x=m+\sqrt{n}$ is a solution to the equation $x^{2}-10 x+1=\sqrt{x}(x+1)$. Find $m+n$.