Processing Math: Done

From Förberedande kurs i matematik 1

Revision as of 08:55, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 2.3:1b moved to Solution 2.3:1b: Robot: moved page) ← Previous diff		Revision as of 10:11, 12 September 2008 (edit) (undo) Ian (Talk | contribs) Next diff →
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-	{{NAVCONTENT_START}}	+	When we complete the square, it is only the first two terms,
-	<center> [[Image:2_3_1b.gif]] </center>	+	<math>x^{2}+2x</math>
-	{{NAVCONTENT_STOP}}	+	, that are involved. The general
		+	formula for completing the square states that
		+	<math>x^{2}+ax</math>
		+	equals
		+
		+
		+	<math>\left( x+\frac{a}{2} \right)^{2}-\left( \frac{a}{2} \right)^{2}</math>
		+
		+
		+	Note how the coefficient
		+	<math>a</math>
		+	in front of the
		+	<math>x</math>
		+	turns up halved in two places.
		+
		+	If we use this formula, we obtain
		+
		+
		+	<math>x^{2}+2x=\left( x+\frac{2}{2} \right)^{2}-\left( \frac{2}{2} \right)^{2}=\left( x+1 \right)^{2}-1</math>
		+
		+
		+	and if we subtract the last "
		+	<math>1</math>
		+	" , we obtain
		+
		+
		+	<math>x^{2}+2x-1=\left( x+1 \right)^{2}-1-1=\left( x+1 \right)^{2}-2</math>
		+
		+
		+	To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,
		+
		+
		+	<math>\left( x+1 \right)^{2}-2=x^{2}+2x+1-2=x^{2}+2x-1</math>
		+
		+
		+	and see that the relation really holds.

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Revision as of 10:11, 12 September 2008

When we complete the square, it is only the first two terms, x2+2x , that are involved. The general formula for completing the square states that x2+ax equals

x+2a2−2a2 

Note how the coefficient a in front of the x turns up halved in two places.

If we use this formula, we obtain

x2+2x=x+222−222=x+12−1 

and if we subtract the last " 1 " , we obtain

x2+2x−1=x+12−1−1=x+12−2 

To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,

x+12−2=x2+2x+1−2=x2+2x−1 

and see that the relation really holds.