Draw the graph of y=3x (4-x) for values of X ranging from -2 to +6 on the same axes draw the line y=5 (x-2) use a scale of 1cm to 1unit on the X axis and 1cm to 5 units on the y axis from your graph
a find the root of the equation 3x (4-x) =0,
b write down the maximum value of y= (4-x),
c deduce the root of the equation 5x (4-x)=5(x-2).
To graph the equation y=3x(4-x) and y=5(x-2) on the same axes, we will first determine some key points and then plot them.
a. The root of the equation 3x(4-x) =0 can be found by setting the expression equal to zero and solving for x:
3x(4-x) = 0
Since a product equals zero when at least one of the factors equals zero, we can set each factor equal to zero:
3x = 0 or 4-x = 0
Solving each equation separately, we find:
x = 0 or x = 4
So the roots of the equation 3x(4-x) =0 are x = 0 and x = 4.
b. The maximum value of y = (4-x) occurs when x is at its minimum value, which is -2 in this case. So:
y = (4-(-2))
y = 6
The maximum value of y= (4-x) is 6.
c. To deduce the root of the equation 5x(4-x) =5(x-2), we can simplify and solve for x:
5x(4-x) = 5(x-2)
Expanding both sides:
20x - 5x^2 = 5x - 10
Rearranging:
5x^2 - 25x + 10 = 0
Dividing both sides by 5:
x^2 - 5x + 2 = 0
We can then use the quadratic formula to find the roots of this equation:
x = (-(-5) ± √((-5)^2 - 4(1)(2))) / (2(1))
Simplifying:
x = (5 ± √(25 - 8))/2
x = (5 ± √17)/2
So the roots of the equation 5x(4-x) =5(x-2) are x = (5 ± √17)/2.