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 Depdendent Variable

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 Dependent Variable

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# Solving Quadratic Equations by Completing the Square

Example

A rectangular patio is 12 feet long and 10 feet wide. We want to increase the length and the width so the area of the new patio will be twice that of the original patio. The length will be increased by twice the amount that the width is increased. By how much should we increase each dimension?

Solution

 Calculate the area of the original patio.  The area of the new patio will be twice that of the original patio. original area   new area = (12 feet)(10 feet) = 120 ft2= 2 Â· 120 ft2 = 240 ft2 Let x = the number of feet the width should be increased. To represent the new width, add x to the original width. Since the length will be increased by twice as much as the width, to represent the new length, add 2x to the original length. new width   new length = 10 + x ft  = 12 + 2x ft The area of the rectangle is: Area = (length)(width) Now, write an equation for the area of the new patio. Substitute the expressions. Multiply the binomials. Simplify. new area  240 240 240 = (new length)(new width)= (12 + 2x)(10 + x) = 120 + 12x + 20x + 2x2 = 120 + 32x + 2x2 This quadratic equation can be solved by completing the square.Step 1 Isolate the x2-term and the x-term on one side of the equation. Subtract 120 from both sides. 120 = 32x + 2x2 Write the equation with decreasing powers of x on the left. 2x2 + 32x = 120 Step 2 If the coefficient of x2 is not 1, divide both sides of the equation by the coefficient of x2. The coefficient of x2 is 2. Divide both sides of the equation by 2. x2 + 16x = 60 Step 3 Find the number that completes the square: Multiply the coefficient of x by . Square the result. The coefficient of the x-term is 16. Step 4 Add the result of Step 3 to both sides of the equation.Add 64 to both sides of the equation. x2 + 16x + 64 = 60 + 64 Step 5 Write the trinomial as the square of a binomial. Write x2 + 16x + 64 as the square of a binomial. Also, simplify the right side of the equation. (x + 8)2 = 124 Step 6 Finish solving using the Square Root Property. Use the Square Root Property. For each equation, subtract 8 from both sides. Step 7 Check each solution. We leave the check for you.

You can simplify the radical:

So the solutions are:

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