Outsmart the Radical Equation Challenge: Can You Crack It?

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Ready for a math challenge? Outsmart the Radical Equation Challenge and see if you can crack it! Dive into this tricky puzzle and put your algebra skills to the test. Perfect for math enthusiasts, students, and anyone looking to sharpen their problem-solving abilities. Can you find the solutions?

In this tutorial, you'll get:

A step-by-step walkthrough of the problem
Key strategies and techniques for solving radical equations
Tips to avoid common mistakes
Insight into advanced problem-solving methods

Think you have what it takes to solve this hard radical math problem? Watch the video, give it a try, and let us know your solution in the comments below.

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@infyGyan

[lukaskamin]

Cool, would be interesting to see how the irrational roots could be checked, is it possible?

[mohammedsaysrashid]

It was an informative and wonderful explanation. Thanks for sharing....finally x=-2

[dorkmania]

After inferring that the equation is self-similar, it simplifies to

x = 2(1 + x)^⅓

Substituting y = x/2 and cubing

y³ = 1 + 2y
=> y³ - 2y - 1 = 0

Using RRT and SDM

(y + 1)(y²-y-1) = 0

Solving

y = -1, (1 ± √5)/2

Or

x = -2, 1 ± √5

[ducduypham]

Divide both sides by 2 we have x/2=cbrt(1+2*cbrt(x+1)). Cube both sides we now have (x/2)^3=1+2*cbrt(x+1). Add both side with x we have Let a=x/2, b=cbrt(x+1) and function f(t)=t^3+2t. The equation will turn into f(a)=f(b). Besides, notice that f(t) is increasing function on R (f'(t)=3t^2+2>0) so a=b or x/2=cbrt(x+1). Multiply both side by 2 and cube both side we have x^3-8x-8=0 <=> (x+2)(x^2-2x-4)=0

[woobjun]

Cubing both sides,
and letting 8x +8 =y^3
results in a system of 2 cub eqns,
x^3 -8 = 8y ..(1)
y^3 -8 = 8x ..(2).
Sutracting
x^3 -y^3 +8(x -y) = 0;
(x -y)(x^2 +xy +y^2) +8(x-y) =0;
(x -y)(x^2 +xy +y^2 +8) =0;
that is,
x -y =0, i.e., x = y ..(3)
Substituting (3) into (1),
x^3 -8x -8 =0,
(x +2)(x^2 -2x -4) =0;
and thus
x = -2
x = [1 +&- sqrt(5)]/2

While
x^2 +xy +y^2 = -8;
4x^2 +4xy +4y^2 = -32;
(2x +1)^2 +3y^2 = -32,
+ve LHS = -ve RHS
yields cmplx sols.

[gnanadesikansenthilnathan]

Could get the solution. Not so challenging