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Solving Equations Containing Radical Expressions










































(√4x + 1) - (√2x +4) = 1

Solve the equation shown for x.

Note that this is not a quadratic equation, even though can be made to look like one mathematically.

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Dec 02, 2011
Solve Equation Containing Radicals
by: Staff


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Part II


This is a quadratic equation. Use the quadratic formula to solve it.

ax² + bx + c = 0

x = [-b ± √(b² - 4ac)]/(2a)

The quadratic formula will yield two solutions. However, remember that there will be only one valid solution.


x² - 48x + 64 = 0

x = unknown

a = 1

b = -48

c = 64


x = {-(-48) ± √[(-48)² - 4*1*64]} / (2*1)


x = [48 ± √(2304 - 256)] / 2

x = [48 ± √(2048)] / 2


x = [48 ± 32√(2)] / 2

x = (48 / 2) ± [32√(2)] / 2

x = 24 ± 16√(2)

x₁ = 24 - 16√(2) ≈ 1.3725830020305

x₂ = 16√(2)+24 ≈ 46.6274169979695


ONLY ONE OF THESE SOLUTIONS IS VALID. The question is: which one, x₁ or x₂ ?



To find the answer, check your work:


For x₁ = 24 - 16√(2) ≈ 1.3725830020305


Substitute 1.3725830020305 for x in the original equation

(√4x + 1) - (√2x +4) = 1

(√(4*1.3725830020305) + 1) - (√(2*1.3725830020305) +4) = 1

(2.3431457505076 + 1) - (1.6568542494924 + 4) = 1

3.3431457505076 - 5.6568542494924 + 4 = 1

1.6862915010152 = 1, NO → x₁ ≈ 1.3725830020305 is NOT a VALID SOLUTION



For x₂ = 16√(2)+24 ≈ 46.6274169979695


Substitute 46.6274169979695 for x in the original equation

(√4x + 1) - (√2x +4) = 1

(√(4*46.6274169979695) + 1) - (√(2*46.6274169979695) +4) = 1

(13.6568542494924 + 1) - (9.6568542494924 + 4) = 1

14.6568542494924 - 13.6568542494924 = 1

1 = 1, YES → x₂ ≈ 46.6274169979695 is a VALID SOLUTION



The final answer is:


x = 16√(2)+24 ≈ 46.6274169979695



Thanks for writing.

Staff
www.solving-math-problems.com



Dec 02, 2011
Solve Equation Containing Radicals
by: Staff


Part I


Question:

How do I solve:

(√4x + 1) - (√2x +4) = 1



Answer:


(√4x + 1) - (√2x + 4) = 1


(a) It’s IMPORTANT TO “PLOT” this particular EQUATION.

(b) It’s IMPORTANT to CHECK the SOLUTION by substituting the value calculated for x into the original equation.


Although the procedure illustrated below shows you how to transform this equation into a quadratic equation to solve it (which yields two solutions), there is ONLY ONE solution.

The equation (√4x + 1) - (√2x + 4) = 1 is not a quadratic equation, even though can be made to look like one mathematically.

This is obvious when looking at a graph of the equation:


Open the following link to see a graph of (√4x + 1) - (√2x + 4) = 1.


(1) If your browser is Firefox, click the following link to VIEW the graph; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page

http://www.solving-math-problems.com/images/graph-01-rad-2011-12-01.png


Solution:



Remove the left parentheses

√4x + 1 - (√2x +4) = 1


Expand the expression in the second parentheses using the distributive law

√4x + 1 - (√2x + 4) = 1


√4x + 1 + (-1) * (√2x + 4) = 1


√4x + 1 + (-1) * √2x + (-1) * 4 = 1


√4x + 1 - √2x - 4 = 1


Take the square root of √4x

√4x + 1 - √2x - 4 = 1

√4 * √x + 1 - √2x - 4 = 1

2 * √x + 1 - √2x - 4 = 1

2√x + 1 - √2x - 4 = 1


Combine like terms (note that 2√x - √2x cannot be combined in their present form)

2√x - √2x + 1 - 4 = 1

2√x - √2x - 3 = 1


Add 3 to each side of the equation

2√x - √2x - 3 + 3 = 1 + 3

2√x - √2x + 0 = 1 + 3

2√x - √2x = 1 + 3

2√x - √2x = 4



Add √2x to each side of the equation

2√x - √2x + √2x = 4 + √2x

2√x + 0 = 4 + √2x

2√x = 4 + √2x

2√x = √2x + 4


Square each side of the equation

(2√x)² = (√2x + 4)²

4x = 2x + 8√(2x) + 16


Subtract 2x from each side of the equation

4x = 2x + 8√(2x) + 16

4x - 2x = 2x - 2x + 8√(2x) + 16

2x = 2x - 2x + 8√(2x) + 16

2x = 0 + 8√(2x) + 16

2x = 8√(2x) + 16


Subtract 16 from each side of the equation

2x = 8√(2x) + 16

2x - 16 = 8√(2x) + 16 - 16

2x - 16 = 8√(2x) + 0

2x - 16 = 8√(2x)


Square each side of the equation again


2x - 16 = 8√(2x)

(2x – 16)² = [8√(2x)]²

4x² - 64x + 256 = 128x


Subtract 128x from each side of the equation

4x² - 64x + 256 = 128x

4x² - 64x + 256 - 128x = 128x - 128x

4x² - 64x + 256 - 128x = 0


Combine like terms


4x² - 64x - 128x + 256 = 0

4x² - 192x + 256 = 0



Divide each side of the equation by 4

4x² - 192x + 256 = 0

(4x² - 192x + 256) / 4 = 0 / 4

(4x² - 192x + 256) / 4 = 0

(4x² / 4) - (192x / 4) + (256 / 4) = 0

x² - 48x + 64 = 0

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