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Completing the Square

by Storm
(Singapore)











































Solve for x by completing the square.

(x+2)ˆ2-7=0

Remember that completing the square is simply a technique for changing the form of a quadratic equation from this:

ax² + bx + c = 0

to this:

(x + z)² - k = 0

Comments for Completing the Square

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Aug 21, 2011
Completing the Square
by: Staff

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Part II

Back to your problem:

(x+2)² - 7 = 0


Add 7 to each side of the equation

(x+2)² - 7 + 7 = 0 + 7

(x+2)² + 0 = 0 + 7

(x+2)² = 7

Take the square root of each side of the equation

√ (x + 2)² = ±√(7)

x + 2 = ±√(7)



Subtract 2 from each side of the equation to remove the 2 from the left side of the equation. This leaves the variable x as the only term on the left side of the equation.

x + 2 - 2 = -2 ±√(7)

x + 0 = -2 ±√(7)

x = -2 ±√(7)


1st value of x₁ = -2 plus the square root of 7

x₁ = -2 + √(7)

x₁ ≈ -2 + 2.6457513110646

x₁ ≈ 0.6457513110646


Notice that the equal sign (=) has been changed to an approximately equal sign (≈).

This is because the solution x₁ = -2 + √(7) is an exact equality. However, x₁ ≈ -2 + 2.6457513110646 is only an approximation. The √(7) is an irrational number which cannot be fully evaluated using the number base of 10. (An irrational number is a number which cannot be written as a fraction. The decimal portion of an irrational number is non-terminating and does not repeat any number sequence.)



2nd value of x₂ = -2 minus the square root of 7

x₂ = -2 - √(7)

x₂ ≈ -2 - 2.6457513110646

x₂ ≈ -4.6457513110646


x ∈{-4.6457513110646, 0.6457513110646}



the final answer is:

x = -2 ±√(7)

or

x ∈{-4.6457513110646, 0.6457513110646}


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http://www.solving-math-problems.com/images/quad-eq-2011-08-21-a.png




check the solution by substituting the two numerical values of x into the original equation


for x₁ ≈ 0.6457513110646


(x+2)² - 7 = 0

(0.6457513110646 + 2)² - 7 ≈ 0

(2.6457513110646)² - 7 ≈ 0

7.00000000000004979038537316 - 7 ≈ 0

.00000000000004979038537316 ≈ 0, OK → x₁ ≈ 0.6457513110646 is a valid solution



for x₂ ≈ -4.6457513110646

(x+2)² - 7 = 0

(-4.6457513110646 + 2)² - 7 ≈ 0

(-2.6457513110646)² - 7 ≈ 0

7.00000000000004979038537316 - 7 ≈ 0

.00000000000004979038537316 ≈ 0, OK → x₂ ≈ -4.6457513110646 is a valid solution




Thanks for writing.

Staff
www.solving-math-problems.com


Aug 21, 2011
Completing the Square
by: Staff


Part I

The question:

by Storm
(Singapore)

(x+2)ˆ2 – 7 = 0


The answer:

(x+2)² - 7 = 0

Most of the work on your problem has already been done.


Completing the square is simply a technique for changing the form of a quadratic equation from this:

ax² + bx + c = 0

to this:

(x + z)² - k = 0


Your equation has already been converted to the proper form:

(x+2)² - 7 = 0


---------------------------------

Background note:

There is a good reason for making this conversion.

Converting a quadratic equation to the new form [(x + z)² - k = 0] makes it easy to solve for x.

Once the quadratic equation has been rewritten in the new form [(x + z)² - k = 0], you simply add “k” to each side of the equation, and then take the square root of each side of the equation. The third and final step is to subtract z from each side of the equation . . . THREE STEPS and you’re finished.

(x + z)² - k = 0

(x + z)² - k + k = 0 + k

(x + z)² = k

√(x + z)² = √k

(x + z) = ±√k

x = -z ±√k

You have probably already recognized that the solution (x = -z ±√k) is, in fact, the quadratic formula:

x = [-b±√(b² - 4ac)]/2a


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