Find the point of intersection.
(-9/100)(x-10)^2+9=(-1/24)(x-12)^2+6
I don't understand how multiplying
y = (-1/2)(x-12)^2 + 6 by 600 would yield
-25(x-12)^2 + 3600 ? Is it supposed to yield
-300x^2+7200x-43,200+3600?
forget about expanding things. You have
(-1/24)A + 6
multiply by 600 to get
-25A + 3600
A is just (x-12)^2
Thank you.
To find the point of intersection of the given equations, let's first solve the equation you provided:
(-9/100)(x-10)^2 + 9 = (-1/24)(x-12)^2 + 6
To simplify the equation, we can multiply through by the least common multiple (LCM) of the denominators of the fractions, which is 1200:
1200 * [(-9/100)*(x-10)^2 + 9] = 1200 * [(-1/24)*(x-12)^2 + 6]
Simplifying the equation further:
-108(x-10)^2 + 10800 = -50(x-12)^2 + 7200
Next, let's distribute the negative sign in both terms:
-108(x^2 - 20x + 100) + 10800 = -50(x^2 - 24x + 144) + 7200
Expanding and simplifying:
-108x^2 + 2160x - 10800 + 10800 = -50x^2 + 1200x - 7200 + 7200
Simplifying the equation even more:
-108x^2 + 2160x = -50x^2 + 1200x
Now, we can move all terms to one side to get a quadratic equation:
-108x^2 + 2160x + 50x^2 - 1200x = 0
Combining like terms:
-58x^2 + 960x = 0
Dividing through by -2 to simplify:
29x^2 - 480x = 0
At this point, we can factor out x to solve for its values:
x(29x - 480) = 0
Setting each factor to zero:
x = 0
29x - 480 = 0
By solving the second equation, we find:
29x = 480
x = 480/29
Thus, the x-coordinate of the point of intersection is x = 0 and x = 480/29.
Now, to find the y-coordinate of the point(s) of intersection, substitute the x-values into either of the original equations. Let's use the first equation:
y = (-9/100)(x-10)^2 + 9
For x = 0:
y = (-9/100)(0-10)^2 + 9
y = (-9/100)(100) + 9
y = -9 + 9
y = 0
So, the point of intersection is (0,0).
For x = 480/29:
y = (-9/100)((480/29)-10)^2 + 9
Now, simplify the equation using a calculator or algebraic manipulation to find the y-coordinate of the second point of intersection.
Regarding your second question about multiplying (-1/2)(x-12)^2 + 6 by 600:
(-1/2)(x-12)^2 + 6 * 600
To distribute the multiplication by 600:
600 * (-1/2)(x-12)^2 + 600 * 6
Multiplying:
-300(x-12)^2 + 3600
So, multiplying (-1/2)(x-12)^2 + 6 by 600 would yield -300(x-12)^2 + 3600, not -300x^2 + 7200x - 43,200 + 3600.