If y^2 -py/2-1=0. Find p
We can solve this quadratic equation by using the quadratic formula:
For a quadratic equation of the form ax^2 + bx + c = 0, the solutions x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is y^2 - py/2 - 1 = 0. So, a = 1, b = -p/2, and c = -1.
Using the quadratic formula, we have:
y = (-(-p/2) ± √((-p/2)^2 - 4(1)(-1))) / (2(1))
Simplifying the equation, we get:
y = (p/2 ± √(p^2/4 + 4)) / 2
This can be further simplified to:
y = (p ± √(p^2 + 16)) / 4
Therefore, the value of p is not a fixed number. It can take on any value, and the solutions for y will depend on the value of p.
If y^2 -py/2-1=0. Find other root
To find the other root, we can apply the quadratic formula as we did before:
y = (p ± √(p^2 + 16)) / 4
To find the other root, we can subtract the root we found from a fixed value, such as p:
Other root = p - (p ± √(p^2 + 16)) / 4
Simplifying this expression, we get:
Other root = (4p - (p ± √(p^2 + 16))) / 4
Expanding and simplifying further, we have:
Other root = (3p ± √(p^2 + 16)) / 4
Therefore, the other root is given by (3p ± √(p^2 + 16)) / 4.
Two places x (56degrees N, 45 degrees E) and y (56degrees N, 32 degrees W) if it takes an aeroplane 6 hours to fly from x to y. Calculate it's speed, correct to the nearest kilometre per hour. (Radius of the earth = 6 40 km, pie = 22/7)
To calculate the speed of the airplane, we need to find the distance between points x and y and divide it by the time it takes to fly from x to y.
First, we need to find the distance (arc length) between the two points on the surface of the Earth.
The distance (arc length) can be calculated using the formula:
Distance = radius of the Earth * angle subtended * π / 180
For point x:
Distance_x = (6400 km) * (45°E - 0°) * (22/7) / 180°
Distance_x = (6400 km) * (45°) * (22/7) / 180°
Distance_x = (6400 km) * (45) * (22/7) / 180
Distance_x ≈ 1803.43 km
For point y:
Distance_y = (6400 km) * (32°W - 0°) * (22/7) / 180°
Distance_y = (6400 km) * (32°) * (22/7) / 180°
Distance_y = (6400 km) * (32) * (22/7) / 180
Distance_y ≈ 1232.06 km
Now, we can find the total distance between x and y:
Total Distance = Distance_x + Distance_y
Total Distance ≈ 1803.43 km + 1232.06 km
Total Distance ≈ 3035.49 km
Finally, we can calculate the speed of the airplane:
Speed = Total Distance / Time
Speed = 3035.49 km / 6 hours
Speed ≈ 505.91 km/h
Therefore, the speed of the airplane is approximately 505.91 km/h.
To find the value of p in the equation y^2 - py/2 - 1 = 0, let's solve this quadratic equation.
Step 1: Rearrange the equation to the standard quadratic form: ay^2 + by + c = 0. In this case, the equation is already in this form.
Step 2: Compare the equation with the standard form, we have a = 1, b = -p/2, and c = -1.
Step 3: Use the quadratic formula to find the value of y:
y = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c into the quadratic formula:
y = (-(-p/2) ± √((-p/2)^2 - 4 * 1 * (-1))) / (2 * 1)
Simplifying further:
y = (p/2 ± √(p^2/4 + 4)) / 2
Step 4: Since the equation is equal to 0, the solutions should be real and equal. Therefore, the discriminant (b^2 - 4ac) must be 0.
(p^2/4 + 4) = 0
p^2 + 16 = 0
Step 5: Solve the equation p^2 + 16 = 0 by subtracting 16 from both sides:
p^2 = -16
Step 6: Taking the square root of both sides, we get:
p = ± √(-16)
Step 7: The square root of a negative number is imaginary. Therefore, the equation has no real solutions for p.
Conclusion: There is no real value of p that satisfies the equation y^2 - py/2 - 1 = 0.
To find the value of p, we can use the quadratic formula.
The general form of a quadratic equation is ax^2 + bx + c = 0. In our case, the equation is y^2 - (p/2)y - 1 = 0.
By comparing this with the general form, we can see that:
a = 1
b = -(p/2)
c = -1
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, we are solving for p, not y. So, we need to rewrite the quadratic formula in terms of p:
y = (-(-p/2) ± √((-p/2)^2 - 4*(1)*(-1))) / (2*(1))
Simplifying this equation further, we have:
y = (p/2 ± √(p^2/4 + 4)) / 2
Now, since we are given the equation y^2 - (p/2)y - 1 = 0, we know that this equation has two identical solutions. This means that the discriminant (the expression inside the square root) must be equal to zero.
So, we set p^2/4 + 4 = 0 and solve for p:
p^2/4 + 4 = 0
p^2/4 = -4
p^2 = -16
p = ± √(-16)
p = ±4i
Therefore, the value of p is ±4i, where i is the imaginary unit.