1. ## Equation of line. Solving quadratic equations.

A straight line passes through the two points A and B with coordinates A(–3,–5) and B(5,1).
a) Determine an equation for the line

Problem 6: roots of quadratic equations
The following quadratic equations have been factorized for you. Read off the roots:
a. x2 – 40x + 300 = (x–10)(x–30) = 0
b. x2 + x – 1 = (x–½)(x+2) = 0
c. x2 – (+)x +  = (x–)(x–) = 0
d. 6x2 + 6x – 72 = (2x–6)(3x+12) = 0
Factorise, and hence determine the roots of the quadratic equation:
e. x2 + 5x + 6 = 0
f. x2 + 6x – 7 = 0

Problem 7: throwing up…
A ball is thrown into the air. The height (over the ground, in metres) of the ball is called y, and a formula for y is y = . Here g denotes the acceleration of gravity, g = 9.8 m/s2, t is the time from when the ball is thrown (in seconds), and v is the speed with which the ball is thrown (in m/s).
a) With your graphing calculator, sketch y as a function of t for a ball thrown upwards with the speed v = 5 m/s. Copy the graph onto paper.
b) With your calculator, find the maximum height the ball reaches (2 sig. figs.)
c) After how many seconds is the ball back on the ground? (2 sig. figs.)

2. 1. The equation for a line can be represented as such:

$\displaystyle y = mx + b$

Where $\displaystyle m$ is the slope and $\displaystyle b$ is the y intercept.

The equation for slope is $\displaystyle \frac {rise}{run}$, or $\displaystyle \frac {y_1-y_2}{x_1-x_2}$. You have two points so you have two x coordinates and two y coordinates. It does not matter what you choose to be $\displaystyle x_1$ so long as you make the corresponding y value $\displaystyle y_1$.

Calculate the slope, then plug it in to the equation $\displaystyle y = mx + b$ using either of your original points to find $\displaystyle b$.

2. When you have an equation like $\displaystyle (x-10)(x-30) = 0$, the roots are whatever makes that equation true. So, for $\displaystyle (x-10)(x-30)$ to be equal to 0, either $\displaystyle (x-10) = 0$, or $\displaystyle (x-30) = 0$.

Now, just solve for $\displaystyle x$ in both cases.

Case 1: $\displaystyle x-10 = 0$, therefore $\displaystyle x = 10$
Case 2: $\displaystyle x-30 = 0$, therefore $\displaystyle x = 30$

So, the roots of the equation are 10 and 30.

3. To factorize a quadratic of the form $\displaystyle x^2 +bx +c = 0$, simply use the sum and product rule. What two numbers have a sum of $\displaystyle b$ and a product of $\displaystyle c$?

$\displaystyle x2 + 5x + 6 = 0$

3 and 2 add up to 5 and multiply to 6, so your factored equation would be:

$\displaystyle (x+2)(x+3) = 0$

4. On your calculator there should be a y= button. Simply enter the equation into it and then press GRAPH. Press 2nd CALC then select MAXIMUM to find the maximum height. Find the two roots of the equations. The time the ball is the absolute value of $\displaystyle x_1 - x_2$

3. Thanks for the help dude