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

 Aug 21, 2011 Completing the Square by: Staff --------------------------------- 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} (1) Click the following link to VIEW the graphical solution; or (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/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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