The graphs of y= x^2 - 8x - 35 and
y= -2(x^2) + 16x +3 intersect in two points. What is the sum of the x-coordinates of the two points of intersection?
Equate the two equation to get:
x^2 - 8x - 35=-2(x^2) + 16x +3
Arrange terms and solve the quadratic equation using the quadratic formula (or otherwise) to get:
x=(-sqrt(258)+12)/3 and
x=(sqrt(258)+12)/3
The sum of the two roots give
24/3 = 8
To find the points of intersection of the two graphs, we need to solve the system of equations formed by setting the two equations equal to each other:
x^2 - 8x - 35 = -2(x^2) + 16x + 3
To simplify the equation, let's distribute -2 on the right side:
x^2 - 8x - 35 = -2x^2 + 16x + 3
Next, let's combine like terms by adding 2x^2 and subtracting 16x from both sides:
3x^2 - 24x - 35 = 3
Now, let's move all the terms to one side to form a quadratic equation:
3x^2 - 24x - 35 - 3 = 0
3x^2 - 24x - 38 = 0
We can now solve this quadratic equation for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -24, and c = -38. Substituting these values into the quadratic formula, we get:
x = (-(-24) ± √((-24)^2 - 4(3)(-38))) / (2(3))
x = (24 ± √(576 + 456)) / 6
x = (24 ± √1032) / 6
Now we have two possible values for x. To find the sum of these values, we need to evaluate each expression and then add them:
x₁ = (24 + √1032) / 6
x₂ = (24 - √1032) / 6
Sum = x₁ + x₂
To get the exact decimal values for x₁ and x₂, you can further simplify the expressions inside the square root and calculate them individually. Once you have the decimal values, add them together to find the sum of the x-coordinates of the points of intersection.