Let r and s be the roots of y^2 - 19y + 7. Find (r-2)(s-2).
To find the value of (r-2)(s-2), we first need to find the values of r and s.
Given that y^2 - 19y + 7 = 0, we can use the quadratic formula to find the roots. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -19, and c = 7. Plugging these values into the quadratic formula, we have:
r = (-(-19) + sqrt((-19)^2 - 4(1)(7))) / (2(1))
r = (19 + sqrt(361 - 28)) / 2
r = (19 + sqrt(333)) / 2
s = (-(-19) - sqrt((-19)^2 - 4(1)(7))) / (2(1))
s = (19 - sqrt(361 - 28)) / 2
s = (19 - sqrt(333)) / 2
Now we can substitute the values of r and s into the expression (r-2)(s-2):
(r-2)(s-2) = ((19 + sqrt(333))/2 - 2)((19 - sqrt(333))/2 - 2)
Simplifying this expression will give us the final answer.
To find the value of (r-2)(s-2), we first need to find the values of r and s, which are the roots of the given quadratic equation.
The given quadratic equation is y^2 - 19y + 7. This equation is in the form of ay^2 + by + c = 0, where a = 1, b = -19, and c = 7.
To find the roots of this equation, we can use the quadratic formula, which states that the roots of the quadratic equation ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -19, and c = 7. Plugging these values into the quadratic formula, we have:
y = (-(-19) ± √((-19)^2 - 4(1)(7))) / (2(1))
This simplifies to:
y = (19 ± √(361 - 28)) / 2
y = (19 ± √333) / 2
So, the roots of the equation y^2 - 19y + 7 are:
r = (19 + √333) / 2
s = (19 - √333) / 2
Now, we can find the value of (r-2)(s-2):
(r-2)(s-2) = [(19 + √333) / 2 - 2][(19 - √333) / 2 - 2]
Simplifying further:
(r-2)(s-2) = [(19 - 4 + √333) / 2][(19 - 4 - √333) / 2]
(r-2)(s-2) = [(15 + √333) / 2][(15 - √333) / 2]
(r-2)(s-2) = (15^2 - (√333)^2) / 4
(r-2)(s-2) = (225 - 333) / 4
(r-2)(s-2) = (-108) / 4
(r-2)(s-2) = -27
Therefore, the value of (r-2)(s-2) is -27.
(r-2)(s-2)
= rs - 2s - 2r + 4
= rs - 2(s+r) + 4
but for y^2 - 19y + 7 , the sum of the roots = 19 and the product of the roots = 7
so ....
rs - 2(s+r) + 4
= 7 - (19) + 4
= -8