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?
(-1/24)(x-12)^2 + 6
consists of two terms, so multiplying by 600 gives me
600 [ (-1/24)(x-12)^2 + 6] , now distribute ...
= -25(x-12)^2 + 3600
= -25(x^2 - 24x + 144) + 3600
= -25x^2 + 600x - 3600 + 3600
= -25x^2 + 600x
( - 9 / 100 ) ( x - 10 ) ^ 2 + 9 = ( - 1 / 24 ) ( x - 12 ) ^ 2 + 6
- 9 * ( x ^ 2 - 2 * x * 10 + 10 ^ 2 ) / 100 + 9 = - 1 * ( x ^ 2 - 2 * x * 12 + 12 ^ 2 ) / 24 + 6
- 9 * ( x ^ 2 - 20 x * 10 + 100 ) / 100 + 9 = - 1 * ( x ^ 2 - 24 x + 144 ) / 24 + 6 Multiply both sides by 100 * 24
- 9 * 100 * 24 ( x ^ 2 - 20 x * 10 + 100 ) / 100 + 9 * 100 * 24 = - 1 * 100 * 24 ( x ^ 2 - 24 x + 144 ) / 24 + 6 * 100 * 24
- 9 * 24 ( x ^ 2 - 20 x * 10 + 100 ) + 9 * 2400 = -1 * 100 * ( x ^ 2 - 24 x + 144 ) + 6 * 2400
- 216 * ( x ^ 2 - 20 x * 10 + 100 ) + 21600 = -100 * ( x ^ 2 - 24 x + 144 ) + 14400
- 216 x ^ 2 + 4320 x - 21600 + 21600 = - 100 x ^ 2 + 2400 x - 14400 + 14400
- 216 x ^ 2 + 4320 x = - 100 x ^ 2 + 2400 x Add ( 100 x ^ 2 - 2400 x ) to both sides
- 216 x ^ 2 + 4320 x + 100 x ^ 2 - 2400 x = - 100 x ^ 2 + 2400 x + 100 x ^ 2 - 2400 x
- 116 x ^ 2 + 1920 x = 0 Divide both sides by 4
- 29 x ^ 2 + 480 x = 0
x ( - 29 x + 480 ) = 0
Solutions :
x = 0
and
- 29 x + 480 = 0 Subtract 480 to both sides
- 29 x + 480 - 480 = 0 - 480
- 29 x = - 480 Divide both sides by - 29
x = - 480 / - 29 = 480 / 29 = 16.551724
Now put x = 0 in equation
( - 9 / 100 ) ( x - 10 ) ^ 2 + 9
OR
( - 1 / 24 ) ( x - 12 ) ^ 2 + 6
all the same
y = 0
Put x = 489 / 29 in equation
( - 9 / 100 ) ( x - 10 ) ^ 2 + 9
OR
( - 1 / 24 ) ( x - 12 ) ^ 2 + 6
all the same
y = 4320 / 841 = 5.136742
Points of intersection:
x = 0 , y = 0
and
x = 489 / 29 , y = 4320 / 841
If you want to see graph go on :
wolframalpha dot com
When page be open in rectangle type :
(-9/100)(x-10)^2+9=(-1/24)(x-12)^2+6
and click option =
To find the point of intersection between the two equations, we can first simplify both equations and then set them equal to each other to solve for x.
Starting with the first equation:
(-9/100)(x-10)^2 + 9 = (-1/24)(x-12)^2 + 6
Let's simplify this equation step by step:
Step 1: Expand the squared terms:
(-9/100)(x^2 - 20x + 100) + 9 = (-1/24)(x^2 - 24x + 144) + 6
Step 2: Distribute the coefficients:
(-9/100)x^2 + (9/100)(20x) - (9/100)(100) + 9 = (-1/24)x^2 + (1/24)(24x) - (1/24)(144) + 6
Simplifying further, we get:
(-9/100)x^2 + (9/5)x - 9 + 9 = (-1/24)x^2 + x - 6 + 6
Negating the fractions:
-(9/100)x^2 + (9/5)x = -(1/24)x^2 + x
Step 3: Move all terms to one side:
-(9/100)x^2 - -(1/24)x^2 + (9/5)x - x = 0
Combining like terms:
-(9/100)x^2 + (1/24)x^2 + (9/5 - 1)x = 0
Step 4: Simplify the equation:
-(33/1200)x^2 + (43/1200)x = 0
Step 5: Multiply both sides by the denominator to clear fractions:
-33x^2 + 43x = 0
Now, let's address the second part of your question about multiplying the equation y = (-1/2)(x-12)^2 + 6 by 600.
If we multiply the equation y = (-1/2)(x-12)^2 + 6 by 600, we get:
600 * y = 600 * [(-1/2)(x-12)^2 + 6]
600y = 600 * (-1/2)(x-12)^2 + 600 * 6
600y = -300(x-12)^2 + 3600
The result is not -25(x-12)^2 + 3600, as there was a mistake in sign during the multiplication.
To clarify, multiplying (-1/2)(x-12)^2 + 6 by 600 would yield -300(x-12)^2 + 3600, not -25(x-12)^2 + 3600.
I hope this helps! Let me know if you have any further questions.