*If the length of a rectangle is 1 unit more than twice the width, and the area is 55 square units, find the length and width.*Solution The formula for the area of a rectangle is Area = Length X Width. At this point, you can see that the solution x = -11/2 is not valid since x represents a measurement of the width and negative numbers are not used for such measurements.

For instance, note that the second form came from adding 7 to both sides of the equation.

Never add something to one side without adding the same thing to the other side.

We can never multiply two numbers and obtain an answer of zero unless at least one of the numbers is zero.

Of course, both of the numbers can be zero since (0)(0) = 0. The solutions can be indicated either by writing x = 6 and x = - 1 or by using set notation and writing , which we read "the solution set for x is 6 and - 1." In this text we will use set notation.

The process of outlining and setting up the problem is the same as taught in chapter 5, but with problems solved by quadratics you must be very careful to check the solutions in the problem itself.

The physical restrictions within the problem can eliminate one or both of the solutions.This can never be true in the real number system and, therefore, we have no real solution.Upon completing this section you should be able to: From your experience in factoring you already realize that not all polynomials are factorable.The solution to an equation is sometimes referred to as the root of the equation.An important theorem, which cannot be proved at the level of this text, states "Every polynomial equation of degree n has exactly n roots." Using this fact tells us that quadratic equations will always have two solutions. The simplest method of solving quadratics is by factoring.Notice that if the c term is missing, you can always factor x from the other terms.This means that in all such equations, zero will be one of the solutions.We will solve the general quadratic equation by the method of completing the square.Certain types of word problems can be solved by quadratic equations.Step 4 Factor the completed square and combine the numbers on the right-hand side of the equation.Step 5 Find the square root of each side of the equation. If step 5 is not possible, then the equation has no real solution. If you can solve this equation, you will have the solution to all quadratic equations.

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