Question

1. solve algebraically:$y=(x - 2)^2$$y=-x^2 - 2x + 1$4) in a circus, a clown throws an apple up off a platform. the height of the apple vs. time can be given by the following equation: $y = -x^2 + 4x + 5$. another clown tries to shoot the apple with a bow and arrow. the height of the arrow vs. time can be given by the following equation: $y = \frac{2}{3}x + 1$. when will they make contact with each other?5) the price of a stock, $a(x)$, over a 12-month period decreased and then increased according to the equation $a(x) = 0.75x^2 - 6x + 20$, where $x$ equals the number of months. the price of another stock, $b(x)$, increased according to the equation $b(x) = 2.5x$ over the same 12-month period. state all prices when both stock values are the same.

Explanation:

Problem 3

Step1: Set equations equal

$(x-2)^2 = -x^2 -2x +1$

Step2: Expand left side

$x^2 -4x +4 = -x^2 -2x +1$

Step3: Bring all terms to left

$x^2 -4x +4 +x^2 +2x -1 = 0$

Step4: Simplify the equation

$2x^2 -2x +3 = 0$

Step5: Calculate discriminant

$\Delta = (-2)^2 -4(2)(3) = 4 -24 = -20$

Step1: Set heights equal

$-x^2 +4x +5 = \frac{2}{3}x +1$

Step2: Eliminate fraction, multiply by 3

$-3x^2 +12x +15 = 2x +3$

Step3: Rearrange to standard form

$-3x^2 +10x +12 = 0$

Step4: Multiply by -1

$3x^2 -10x -12 = 0$

Step5: Apply quadratic formula

$x = \frac{10\pm\sqrt{(-10)^2 -4(3)(-12)}}{2(3)} = \frac{10\pm\sqrt{100+144}}{6} = \frac{10\pm\sqrt{244}}{6} = \frac{10\pm2\sqrt{61}}{6} = \frac{5\pm\sqrt{61}}{3}$

Step6: Keep positive time

$x = \frac{5+\sqrt{61}}{3} \approx 4.47$

Step1: Set stock prices equal

$0.75x^2 -6x +20 = 2.5x$

Step2: Rearrange to standard form

$0.75x^2 -8.5x +20 = 0$

Step3: Eliminate decimals, multiply by 4

$3x^2 -34x +80 = 0$

Step4: Factor the quadratic

$(3x -10)(x -8) = 0$

Step5: Solve for x

$x = \frac{10}{3} \approx 3.33$ or $x=8$

Step6: Find prices with $B(x)$

For $x=\frac{10}{3}$: $B(\frac{10}{3})=2.5\times\frac{10}{3}=\frac{25}{3}\approx8.33$
For $x=8$: $B(8)=2.5\times8=20$

Answer:

There are no real solutions; the complex solutions are $x = \frac{1\pm i\sqrt{5}}{2}$

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Problem 4