The solution of a quadratic equation is also called the root(s) of the equation. In the cases we will be looking at, solving an equation involves factoring (for equations that cannot be factored, see the next unit).
The steps we will need to follow are as follows:
1. Arrange the quadratic equation in the form ax² + bx + c = 0.
2. Factor the expression.
3. Set each factor to zero.
4. Solve for each factor.
Example 1 Solve x² - x = 6.
x² - x - 6 = 0
(x + 2)(x - 3) = 0
(x + 2) = 0 or (x - 3) = 0
x = -2, or x = 3
Thus, the solutions, or roots, are -2 and 3. NOTE: This is where the parabola crosses the x axis.
Example 2: A soccer ball is kicked into the air at a speed of 20 m/s. Its height, h, in meters, after t seconds is given by the formula:
h = -5t² + 20t
a) How long after it is kicked is the soccer ball at a height of 15 m?
b) For how long is the soccer ball above 15 m?
a) When the soccer ball is 15 m high, h = 15.
Substitute 15 for h in the formula h = -5t² + 20t.
15 = -5t² + 20t
Collect all the terms on one side of the equation.
5t² - 20t + 15 = 0
Remove a common factor of 5 from each term:
5(t² - 4t + 3) = 0
Divide each side by 5.
t² - 4t + 3 = 0
Factor.
(t - 1)( t -3) = 0
t - 1 = 0 or t -3 = 0
t = 1 or t = 3
Therefore, the soccer ball is at a height of 15 m twice: once on its way up, at 1 s, and once on its way down, at 3 s after the kick.
b) The ball is above 15 m when t is greater than 1 s, and less than 3 s. Therefore, for 3 s - 1 s = 2 s.
Thus, the ball is above 15 m for 2 s.
Practice Questions
Page 315-16 #1 abfh, 2bfjl, 3a, 9, 13.
