TenMarks teaches you how to solve real life problems on quadratic equations by completing the square.
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Learn Applications of Solving Quadratic Equations by Completing the Square Word problems. So the problem it states the Raven wants to create a rectangular garden in her backyard. She wants it to have a total area of 120 feet and it should be 12 feet longer than it is wide. What dimensions should she use for the garden? Round to the nearest hundredth of a foot. All right, so we don’t know the width of the garden. So we’re going to say lets the width be x. So we know that the length of the garden is x+12 feet because it should be 12 feet longer than it is wide. So the length of the garden is 12 feet longer than it is wide. All right, and we also know that the area of the garden is 120 square feet. So to find the area, we know that the formula for an area is L×W. So we’re going to go ahead and make our substitutions. So we know the area is 120 feet and we know that our length is whatever the width is plus 12, so our length is x+12. And then, we multiply that by our width which we said was x. So our width is x. So, now we go ahead and we’re going to multiply. So 120=x×x is x2+x×12 is 12x. So 120=x12+12x or we can rewrite it and say x2+12x=120. All right, now here are the steps to find x. Now we need to solve for x. So our first step is we need to make sure and write the equation in the form x2+bx=c. however, our equation is already written in that form. So we don’t have to worry about changing it into this form, so we’re okay with this. So our second step then is we need to add (b/2)2 to both sides of the equation. So in this equation b is 12, so b=12. So I'm going to take (12÷2)2 which gives me 62 which gives me 36. So that means I'm going to add 36 to both sides of the equation. So that means I get x2+12x+36=120+36, so now I’ve added 36 to both sides of the equation. So I get x2+12x+36 and that would give me 156. Now our third step is we factor and simplify. So, x2+12x+36 is written in the form, so its written in the form x2+bx+(b/2)2, so its written in this form. And we know that x2+bx+(b/2)2 is equal to x+(b/2)2. So we know that there equal to each other. So that means that x+b/2, b is 12. So 12/2 would be 6, so we know that x+62 is equal to 156. So now we need to take the square root of both sides. So I take the square root of both sides, it would be x+6=v156 and -v156. So if I take the approximate, so the approximate square root would be of 156 would be 12.489 and -12.489. So now, we need to write and solve the two equations. So now we’re going to write and solve our two equations. So here, we have x+6=12.489 and we have x+6=-12.489. So that means if I subtract 6 from both sides, I get x=6.489 or x=-18.489. So remember the width cannot be a negative. You can't have a negative width, so it cannot be a negative. So we can only use the positive number and in our word problem if you remember. The word problem said we need to round to the nearest hundredth of a foot. So if I take x=6.489 and I round to the nearest hundredth, I get 6.49. So that means that the width would be x which equals 6.49 and remember that our length of the garden then was our width plus 12. So our length would be x+12, so that would be 6.49+12 which is equal to 18.49 feet. So the dimensions of the garden would be 6.49 feet by 18.49 feet. All right, so things to keep in mind that when a trinomial is a perfect square there is a relationship between the coefficient of the x-term and the constant term. So if x+n2 is equal to x2+2nx+n2 or and (2n/2)2=n2 and the same could be true with (x-n)2, and that would be equal to x2-2nx+n2. And that would be also equal to (-2n÷2)2 which is equal to n2. And then remember to complete the square of x2+bx. You add (b/2)2 to the expression and this will form a perfect square trinomial.
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