1. The sum of the squares of two numbers is 74 and the difference of their squares is 24.
a) What system of equations models this situation?
b) What are the two numbers?
2. For what value of b will the line y= -2x+b be tangent to the parabola y=3x^2+4x-1?
(x+y)(x-y) = 24
x^2+y^2 = 74
you want some place where the tangent to the parabola has slope = -2
6x+4 = -2
x = -1
so, y=-2
The tangent line at (-1,-2) is
y+2 = -2(x+1)
y = -2x-4
b = -4
see
http://www.wolframalpha.com/input/?i=plot+y%3D3x^2%2B4x-1%2C+y+%3D+-2x-4
1. a) To model this situation with a system of equations, we can let the two numbers be represented by variables, such as x and y.
Let x represent the first number and y represent the second number. The sum of their squares is 74, so we have the equation:
x^2 + y^2 = 74
The difference of their squares is 24, so we have the equation:
x^2 - y^2 = 24
These two equations form the system that models the situation.
b) To find the values of x and y, we can solve the system of equations:
x^2 + y^2 = 74 (Equation 1)
x^2 - y^2 = 24 (Equation 2)
We can solve this system by using the method of elimination. Adding Equation 2 to Equation 1 eliminates the y^2 terms:
(x^2 + y^2) + (x^2 - y^2) = 74 + 24
2x^2 = 98
Dividing both sides by 2 gives:
x^2 = 49
Taking the square root of both sides gives:
x = ± 7
Substituting this value of x into Equation 1, we can find the value of y:
7^2 + y^2 = 74
49 + y^2 = 74
y^2 = 25
y = ± 5
Therefore, the two numbers can be (7, 5) or (-7, -5).
2. To find the value of b for which the line y = -2x + b is tangent to the parabola y = 3x^2 + 4x - 1, we need to find the point of tangency.
Let's equate the two equations:
-2x + b = 3x^2 + 4x - 1
Rearranging the equation in standard form:
3x^2 + 6x - 2x + 4x - b + 1 = 0
Combining like terms:
3x^2 + 8x - b + 1 = 0
To determine when the line is tangent to the parabola, the quadratic equation must have exactly one solution. This means that the discriminant must be zero.
The discriminant formula is:
b^2 - 4ac
In this case, a = 3, b = 8, and c = -b + 1:
8^2 - 4(3)(-b + 1) = 0
64 + 12(b - 1) = 0
64 + 12b - 12 = 0
12b + 52 = 0
12b = -52
b = -52/12
Simplifying, we find:
b = -13/3
Therefore, the value of b that makes the line y = -2x + b tangent to the parabola y = 3x^2 + 4x - 1 is b = -13/3.
1. To model this situation with a system of equations, we can let the two numbers be represented by variables. Let's call them x and y.
a) The sum of the squares of two numbers is 74. This can be represented by the equation:
x^2 + y^2 = 74
b) The difference of their squares is 24. This can be represented by the equation:
x^2 - y^2 = 24
To solve this system of equations, there are several methods you can use, such as substitution or elimination. Let's use substitution in this case.
From equation 2, we can solve for x^2:
x^2 = y^2 + 24
Substitute this value of x^2 into equation 1:
(y^2 + 24) + y^2 = 74
2y^2 + 24 = 74
2y^2 = 50
y^2 = 25
y = ±5
Substituting the value of y back into equation 1, we can find the corresponding values of x:
x^2 + (±5)^2 = 74
x^2 + 25 = 74
x^2 = 49
x = ±7
So, the two numbers are (7, 5) and (-7, -5).
2. To find the value of b that makes the line y = -2x + b tangent to the parabola y = 3x^2 + 4x - 1, we need to find the point where the line and the parabola intersect.
Set the two equations equal to each other:
-2x + b = 3x^2 + 4x - 1
Rearrange the equation to quadratic form:
3x^2 + 6x + b - 4x + 1 = 0
3x^2 + 2x + (b + 1) = 0
For the line to be tangent to the parabola, it means that there should be only one real solution for this equation. In other words, the discriminant, b^2 - 4ac, should be equal to 0.
The discriminant is:
(2^2) - (4)(3)(b + 1)
Setting the discriminant equal to 0:
4 - 12(b + 1) = 0
-12b - 8 = 0
-12b = 8
b = -8/12
b = -2/3
So, the value of b that will make the line y = -2x + b tangent to the parabola y = 3x^2 + 4x - 1 is b = -2/3.