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Completing the Square
Initial StepGeometric Demonstration of Identity (x+Q)2 = x2+2Qx + Q2. valid for x and Q positive follows. Draw a square with sides of length x+Q and then divide the sides into sub-segments of length x and Q respectively as indicated below.
Here the equality of two different ways to calculate area of square yields
the require identity.
The identify (x+Q)2 = x2+2Qx + Q2 also follows in general from the earlier identity
which we established earlier if we take A = B = Q in it. 2. Completing the square identity:x2+2Qx +P= (x+Q)2 - Q2 + P is consequence of the identity (x+Q)2 = x2+2Qx + Q2
Variation:x2-2Qx +P = (x-Q)2 - Q2 + P is consequence of the identity (x-Q)2 = x2 -2Qx + Q2 3. ExamplesExample Ix2+6x + 5 =
Example IIx2-8x + 25 =
Example IIIx2-8x + 12 =
Example IVx2-10x + 8 =
Note: Completing the square may lead to a difference of squares or a sum of squares. In the first case, how to factor the difference of squares leads to the factorization of quadratic expressions and to the solution of quadratic equations. See below. Lessons on Quadratics: | ||||||||||||||||
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Road
Safety Message Do not walk on a road with your back to the
traffic - rule of thumb
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