Answer:
y = (1/4)x² - (5/4)x + 1
Step-by-step explanation:
The x-intercepts of the quadratic equation are simply it's roots.
Thus, we have;
(x + 1) = 0 and (x - 4) = 0
Now, formula for quadratic equation is;
y = ax² + bx + c
Where c is the y intercept.
At y-intercept: (0,1), we have;
At (-1,0), thus;
0 = a(1²) + b(1) + 1
a + b = -1 - - - (1)
At (4,0), thus;
0 = a(4²) + b(4) + 1
16a + 4b = -1
Divide both sides by 4 to get;
4a + b = -1/4 - - - (2)
From eq 1, b = -1 - a
Thus;
4a + (-1 - a) = -1/4
4a - 1 - a = -1/4
3a - 1 = -1/4
3a = 1 - 1/4
3a = 3/4
a = 1/4
b = -1 - 1/4
b = -5/4
Thus;
y = (1/4)x² - (5/4)x + 1
Answer:
z = -12
Step-by-step explanation:
The given system of equations is:
xy/(x + y) = 1 ...........................(1)
xz/(x + z) = 2...........................(2)
yz/(y + z) = 3...........................(3)
From (1): x + y = xy
=> y = xy - x
y = x(y - 1)
x = y/(y - 1).......................................(4)
From (2): 2(x + z) = xz
=> 2x + 2z = xz
2x = xz - 2z
2x = z(x - 2)
z = 2x/(x - 2) ....................................(5)
From (3): 3(y + z) = yz
=> 3y + 3z = yz
3y = yz - 3z
3y = z(y - 3)
z = 3y/(y - 3)....................................(6)
Comparing (5) and (6)
2x/(x - 2) = 3y/(y - 3)
2x(y - 3) = 3y(x - 2)
2xy - 6x = 3xy - 6y
6(y - x) = xy .................................(7)
But from (1): xy = x + y
Using this in (7), we have
6(y - x) = x + y
6y - y - 6x - x = 0
5y - 7x = 0
5y = 7x
x = 5y/7................................................(8)
Using this in (4)
5y/7 = y/(y - 1)
1/(y - 1) = 5/7
(y - 1) = 7/5
y = 1 + 7/5
y = 12/5..........................................(9)
Using this in (8)
x = 5(12/5)/7 = 12/7 .......................(10)
Using (10) in (5)
z = 2x/(x - 2)
z = 2(12/7) ÷ (12/7 - 2)
= 24/7 ÷ -2/7
= 24/7 × (-7/2)
= -24/2 = -12
z = -12.
We are given that the angle a is the right angle. So let
us work from this.
ab = 12 (the vertical side of the triangle)
bc = 13 (which if drawn can be clearly observed to be the
hypotenuse) = the side opposite to angle a
ca = 5 (the horizontal side of the triangle)
Since we are to find for the cosine ratio of angle c or
angle θ, therefore:
cos θ = adjacent side / hypotenuse
cos θ = ca / bc
cos θ = 5 / 13
Check out the attached image below for the illustration
of the triangle.
Answer:
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