SOLUTION: Can someone please help me?????? 1) Number Problem: Find two consecutive positive integers such that the sum of their squares is 85. 2) 3x^2-2x+1=0 3) Constructio

Question 79590: Can someone please help me??????
1) Number Problem: Find two consecutive positive integers such that the sum of their squares is 85.

2) 3x^2-2x+1=0

3) Construction: A garden area is 30 ft. long and 20 ft. wide. A path of uniform width is set around the edge. If the remaining garden area is 400 ft^2, what is the width of the path?
Thanks!:)
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
1) Number Problem: Find two consecutive positive integers such that the sum of their squares is 85.
:
The two consecutive numbers:
x, (x+1)
:
Sum of their squares = 85:
x^2 + (x+1)^2 = 85
:
x^2 + x^2 + 2x + 1 = 85; FOILed (x+1)(x+1)
:
2x^2 + 2x + 1 - 85 = 0
:
2x^2 + 2x - 84 = 0; a quadratic equation
:
x^2 + x - 42; simplified, divided equation by 2
:
(x + 7)(x - 6) = 0; easily factored
:
x = +6 or x = -7; either solution will work
:
numbers would be +6, +7 or -7, -6
:
Check: 6^2 + 7^2 = 85 or -7^2 + -6^2 = + 85 also
:
:
:
2) 3x^2-2x+1=0
This equation has no real roots. Look at the discriminant (b^2 - 4*a*c)
4 - 4*3*1; a negative value
:
:
3) Construction: A garden area is 30 ft. long and 20 ft. wide. A path of uniform width is set around the edge. If the remaining garden area is 400 ft^2, what is the width of the path?
:
Draw a diagram of this. Label the path width as x;
Label the rectangle 30 by 20, it will be apparent to you that the inner rectangle which is the garden, dimension will be (30-2x) by (20-2x)
That area is given as 400 sq/ft. So we have:
:
(20-2x)*(30-2x) = 400
FOIL
600 - 100x + 4x^2 = 400
:
Arrange as a quadratic:
4x^2 - 100x + 600 - 400 = 0
4x^2 - 100x + 200 = 0
:
Simplify, divide equation by 4
x^2 - 25x + 50 = 0
:
Use the quadratic formula to find x: a=1; b= -25; c=50
I assume you know how to do that:
I got approx values of x = 22.81 and x = 2.19, obviously it's the smaller value
:
check this: 2x = 4.38, subtract that from 30 and 20 and get the dimension of the garden, find the area and see if it is, in fact, 400 sq/ft:
25.62 * 15.62 = 400.2 ~ 400 sq ft
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