forty-two 2 years ago Margaret is planning a rectangular garden. Its length is 4 feet less than twice its width. Its area is 170ft^2. what are the dimensions of the garden?

1. mathstudent55

The area of a rectangle is A = LW, where A = area, L = length, W = width The width is W The length is 4 feet less than twice the width twice the width = 2W four feet less than twice the width = 2W - 4 L = 2W - 4 A = LW, but A = 170, so 170 = LW L = 2W - 4 Since the second equation is already solved for L, just plug that into the first equation and solve for W. Then plug in W in the second equation and solve for L

2. forty-two

170=(2w-4)w once you get here do you distribute?

3. forty-two

@mathstudent55

4. mathstudent55

yes, distribute

5. forty-two

170=(2w-4)w 170=2w^2-4w 0=2w^2-4w-170 I dont think im doing this right

6. mathstudent55

It looks good to me.

7. mathstudent55

Now try to factor the right side.

8. forty-two

can you help me with that, im not sure how

9. mathstudent55

First, divide both sides by 2

10. mathstudent55

also, switch sides

11. mathstudent55

2w^2 - 4w - 170 = 0 Divide both sides by 2

12. forty-two

okay so 2w^2-4w-170=0 w^2-2w-85=0?

13. mathstudent55

Right. Now this kind of factoring involves simply finding two numbers that multiply to -85 and add to -2

14. mathstudent55

-85 = -85 x 1 -85 + 1 = -84 -85 = 85 x -1 85 - 1 = 84 -85 = -17 x 5 -17 + 5 = -12 -85 = 17 x (-5) 17 - 5 = 12

15. mathstudent55

As you can see, there are no two such numbers. This cannot be factored. We now use the quadratic equation. Let's move to a drawing.

16. mathstudent55

|dw:1356497065032:dw|

17. mathstudent55

That is the quadratic formula. If you have an equation of the form ax^2 + bx + c = 0, you enter the coefficients a, b, and c into the quadratic formula, as shown above, and you get the solution to the equation.

18. mathstudent55

In your case we are solving for w instead of x, and we have a = 1, b = -2, and c = -85.

19. mathstudent55

|dw:1356497319940:dw|

20. forty-two

okay so is the answer w = 2SQRT(86)?

21. Raja99

longest ans!

22. mathstudent55

|dw:1356498098987:dw|

23. forty-two

i know, Im so sorry for mathstudent55

24. mathstudent55

Don't be sorry. this is fun for me.

25. forty-two

Okay :) thank you so much that was a lot of help

26. mathstudent55

I wrote x in the drawing but I meant W. Now you know that W = 1 + sqrt(86) or W = 1 - sqrt(86)

27. forty-two

okay, thank you sooooo much

28. mathstudent55

First look at 1 - sqrt(86). It's a negative number. A width can't be negative, so discard that solution. W = 1 + sqrt(86) L = 2W - 4 = 2(1 + sqrt(86)) - 4 = 2 + 2sqrt(86) - 4 = 2sqrt(86) - 2 So final answer is: L = 2sqrt(86) - 2 W = 1 + sqrt(86)

29. mathstudent55

You're welcome.