# Lesson 17

• Let’s explore the meaning of quadratics.

### 17.1: Area Between Triangles

The area of the shaded region from the image can be represented by the expression $$\frac{1}{2}(2+2a)(2+2a) - \frac{1}{2}\boldcdot2^2$$ which can be rearranged to $$2a^2 + 4a$$. To find the value of $$a$$ when the shaded area is 30 square centimeters, Mai sets up the equation $$2a^2 + 4a = 30$$.

1. One solution to the equation is $$a = \text{-}5$$. Find another solution. Explain or show your reasoning.
2. What do the 2 solutions to the equation represent in this situation? Do the values make sense?

### 17.2: Getting the Ball Off the Roof

A ball is kicked off the roof of a building so that its height above the ground, given in feet, $$t$$ seconds after it is kicked is represented by the equation $$h(t) = \text{-}16t^2 + 33t + 37$$.

1. At what height is the ball when it is kicked? Explain or show your reasoning.
2. At what height is the ball 2 seconds after it is kicked? Explain or show your reasoning.
3. What does it mean for the situation when $$h(t) = 8$$?
4. What does it mean for the situation when $$t = 1.3$$?
5. Graph the function.
1. Approximate the number of seconds after the ball is kicked when it will hit the ground. Explain how you know.
2. Approximate the number of seconds after the ball is kicked when it will reach its highest point. Explain how you know.
3. Approximate the number of seconds after the ball is kicked when it will reach its starting height again. Explain how you know.
6. Write an equation that represents the exact moment when the ball hits the ground.

### 17.3: Kicking the Field Goal

Andre kicks a football for a field goal. The height above ground, given in feet, $$t$$ seconds after it is kicked, is represented by the equation $$g(t)=\text{-}16t^2+56t+0.5$$.

1. At what height is the ball when it is kicked? Explain or show your reasoning.
2. At what height is the ball 2 seconds after it is kicked?
3. What does it mean for the situation when $$g(t)=10$$?
4. What does it mean for the situation when $$t=1.7$$?
5. Graph the function.
1. Approximate the number of seconds after the ball is kicked when it will hit the ground. Explain how you know.
2. Approximate the number of seconds after the ball is kicked when it will reach its highest point. Explain how you know.
3. Approximate the number of seconds after the ball is kicked when it will be 10 feet above the ground for the second time. Explain how you know.
6. Write an equation that would give the exact time when the ball is 10 feet above the ground.