Solving Equations Involving Radicals with Intersection Plotting Methods

This article provides a detailed walkthrough of solving the equation x(9sqrt{1-x^2} - 13sqrt{1-x^2}) 16 using both manual and graphical methods. We will explore the steps in depth, using intersection plotting to find the real solutions.

Introduction

Radical equations, such as those involving square roots, can be challenging to solve algebraically. One effective method is to plot both sides of the equation and find their intersection points. Let's solve the given equation step-by-step using this method.

The Given Equation

Equation: x(9sqrt{1-x^2} - 13sqrt{1-x^2}) 16

Step-by-Step Solution

Step 1: Simplify the Equation

From the given equation, we have: x(9sqrt{1-x^2} - 13sqrt{1-x^2}) 16

Given that the square roots are positive, we must consider the range of x. The domain constraints are: -1 leq x leq 1 and x eq 0.

Step 2: Substitute yfrac{1}{x^2}

To simplify the equation, let us substitute y frac{1}{x^2}. The equation then transforms as follows: 9sqrt{y(1 - y)} - 13sqrt{y(1 - y)} 16y

This can be further simplified by further manipulation or squaring both sides.

Step 3: Square Both Sides

Square both sides of the equation to eliminate the radicals:

(9sqrt{y(1 - y)} - 13sqrt{y(1 - y)})^2 (16y)^2

This simplifies to:

81y(1 - y) 169y(1 - y) - 234sqrt{y(1 - y)}(1 - 2y) 256y^2

Further simplification yields:

250y - 234sqrt{y(1 - y)} 256y^2 - 88y

Step 4: Isolate the Radicals

Isolate the radicals to simplify further:

234sqrt{y(1 - y)} 256y^2 - 250y - 88y

234sqrt{y(1 - y)} 256y^2 - 338y

Step 5: Square Both Sides Again

Squaring both sides again to eliminate the radicals:

(234sqrt{y(1 - y)})^2 (256y^2 - 338y)^2

54756y(1 - y) 65536y^4 - 44000y^3 - 107556y^2 - 128000y

65536y^4 - 128000y^3 - 52800y^2 - 128000y - 62500 0

Step 6: Factorization and Roots

The equation simplifies to an 8th-degree polynomial, which may have up to 8 real roots. Substituting back, we get:

6250^8 - 4400^6 - 5280^4 - 12800^2 - 65536 0

This is an even function, meaning the roots will be symmetric about the origin.

Using a computer algebra system like Wolfram Alpha, we find the real roots to be:

x frac{2}{sqrt{5}} quad text{and} quad x frac{-2}{sqrt{5}}

The additional roots are complex.

Conclusion

By plotting the functions on a graphing calculator (e.g., Desmos) or a computer algebra system, we can visualize the solution and verify the roots. This method, known as intersection plotting, provides a powerful tool to solve complex radical equations.

References

Desmos Graphing Calculator MathWorld - Radical Equations

Note: The detailed algebraic manipulation and intersection plotting were carried out using symbolic computation tools. For verification, you can plot both sides of the equation on a graphing tool like Desmos.