Write a quadratic equation having the given number as solutions:
c/2 and d/2
please help me solve
I will use x as my variable
if x = c/2 then 2x-c = 0
if x = d/2 then 2x-d = 0
your quadratic equation would be
(2x-c)(2x-d) = 0
expand it to get it into general form
OR
sum of roots = c/2 + d/2 = (c+d)/2
product of roots = cd/4
equation: x^2 - (c+d)/2 x + cd/4 = 0 or
4x^2 - (c+d)x + cd = 0
last line should read:
4x^2 - 2(c+d)x + cd = 0
To write a quadratic equation with c/2 and d/2 as solutions, we can use the fact that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions can be found using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Given that the solutions are c/2 and d/2, we can equate them to x:
c/2 = (-b ± √(b^2 - 4ac)) / 2a ...............(1)
d/2 = (-b ± √(b^2 - 4ac)) / 2a ...............(2)
Now, we can solve the above two equations simultaneously to find the values of a, b, and c.
First, let's solve equation (1) for c/2:
c/2 = (-b ± √(b^2 - 4ac)) / 2a
Multiply both sides by 2:
c = -2b ± 2√(b^2 - 4ac) ...............(3)
Next, let's solve equation (2) for d/2:
d/2 = (-b ± √(b^2 - 4ac)) / 2a
Multiply both sides by 2:
d = -2b ± 2√(b^2 - 4ac) ...............(4)
Comparing equations (3) and (4), we can see that they are identical. Therefore, we can conclude that c = d.
Now, let's simplify equations (3) and (4) by removing the ± symbol (since we know that c = d):
c = -2b + 2√(b^2 - 4ac) ...............(5)
c = -2b - 2√(b^2 - 4ac) ...............(6)
We can combine equations (5) and (6) to eliminate the √ symbol:
-2b + 2√(b^2 - 4ac) = -2b - 2√(b^2 - 4ac)
Rearrange the terms:
4√(b^2 - 4ac) = 0
Divide both sides by 4:
√(b^2 - 4ac) = 0
Square both sides to remove the square root:
b^2 - 4ac = 0 ...............(7)
Equation (7) is the quadratic equation with solutions c/2 and d/2.